Thanks for joining us. I'm Peter Bergen, Vice President at New America. And we are here to talk about this great new book by Laura Tillman, The Migrant Chef, The Life and Times of Lalo Garcia. It's just a kind of some housekeeping notes. There's in the chat, there will be links to buy the book. And then also, as the announcer said, if you have questions, put it in Slido and I'll moderate them and ask Laura those questions as they come in. So let me first of all introduce Laura. She is a colleague of mine at the podcast that we produced for Audible in the room. She's also published another book, The Long Shadow of Small Ghosts. She began her career as a newspaper reporter on the US Mexico border in 2007 before moving to Mexico City, where she is right now in 2014. She had her writings appeared in New York Times, Los Angeles Times, then the nation. She's also a member of the faculty at Gallagher College where she teaches and she lives in Mexico City with her family. So Laura, over to you. Thank you, Peter. As Peter said, we work on this great podcast that you should also listen to if you're tuning in in the room with Peter Bergen. And it was just really generous of Peter to do this event with me for the migrant chef. You know, in my role on the podcast, I think a lot about narrative structure and how we can communicate complex ideas through storytelling. We actually have an episode coming out about that theme tomorrow about how you sort of turn the news into great writing and writing that people actually want to read. So that's something that I think about a lot. And as someone who thinks a lot about narrative, I'm a big believer in chronology as a a helpful tool to bring people along. And so I figured I would sort of use chronology in this case as my as my friend to just begin at the beginning and tell you a little bit about me and how I came to this story in the first place. So I moved from New York to South Texas in 2007. And there I got my first job in a border city, Brownsville, Texas at the newspaper. I was freelancing for a few years. And then I began in a low residency MFA program. And in that program, I started writing a thesis that would become my first book, which was about a death penalty case. I came across that case not because I was interested in writing about murder or crime, but because there was this building near the newspaper office that the city wanted to tear down. And it was a really dilapidated building, kind of a rooming house where there were you could call them apartments, but really it was just rooms with communal bathrooms. And this murder had taken place a few years earlier. And there had been a trial and one of the one of the people in the couple had been sent to death row. And this building just seemed like kind of a stand in for this crime. And I think one of the things that that project taught me, which I brought into this project is kind of the value of following a subject over the long term, and also the value of kind of thinking about subjects outside of the box of what the way those stories are often told. So instead of approaching this case of these murders as a kind of true crime story about the crime itself, I approached it kind of through the lens of this building and the fate of this building and what this building revealed about the community and its relationship to the crime and the process of healing from tragedy. And I spent six years following that story. And I spent a lot of time just kind of immersing myself in this neighborhood around this building with these people, trying to learn from them, trying to hear how those stories and narratives changed subtly over the years. I went to death row, I met this man who had committed the crime. And I started to think a lot about this idea of erasure versus remembrance when it comes to tragic events. And so, you know, that kind of brings me to in 2016, that book was published. And now I am the mother of two human babies, but that was my first book, baby. And I had this kind of strong feeling of postpartum, you know, depression after my book came out in terms of just feeling the sense of loss that I had had this project that for many years, as a freelancer at the time, had given me a lot of meaning in my work. And I was, you know, working on things as a freelancer, I think sometimes you're working on things that are really compelling. And sometimes things that are more for income. And when I had this book project I was working on, I had this feeling that I was doing something that in the background gave me a lot of meaning in my work. And so I started looking around for another project. And this time I wasn't actually looking for something that was supposed to be long term, I was looking for more of a short term. I thought that I had kind of, you know, I'd been through the MFA program and I had gotten to the point where maybe in a year I could immerse myself in a subject and write a book. And so I was in Mexico City, I had recently moved there. By the way, I'm here in Mexico City. And if you at any point here, kind of the street sounds of Mexico City, I apologize, but they're quite loud and it's kind of hard to insulate oneself from them sometimes. But basically I, you know, I think that one of the ways that a lot of people get to know a foreign culture, and in particular in Mexico is through the food. And so I was living here, I was reporting, I was doing some assignments for Los Angeles Times, and I was eating a lot of amazing food in these restaurants in Mexico City. And there was this kind of electricity to the dining scene here at that moment where, you know, it felt like there were all these really interesting connections that were being drawn between different cultures, both, you know, Italian and Mexican food, French and Mexican food, Indian and Mexican food, but also modern and pre-Hispanic cuisines. And it felt like a kind of intellectual and creative conclusion of a much longer story. It didn't feel like just a moment in time in a food scene where there was kind of a gimmick. And so I think that one of the things that if you, you know, if you're someone who has eaten in fine dining restaurants today or even watched shows like Top Chef, you might have a sense that narrative has become a big part of dining culture as well, that there are a lot of narratives that are told in restaurants in the front of house about food. And at times that's about, you know, the story of the protagonist in the kitchen, the chef. Sometimes that's more of a cultural story or a story about sustainability. You know, in Mexico City, for example, you might see on a menu that some of the food that you were eating was grown in a chinampa. And so you might think, well, you know, what's a chinampa? And that kind of opens up this conversation about pre-Hispanic Mexico, that Mexico, when it was Tenochtitlan, was an island city on a lake. And Zochimilco in the south of Mexico City is the only part of the city today that still resembles what that pre-Hispanic city might have looked like. It's a city with all of these canals and waterways. And there are these islands that since, you know, the time of the Zochimiltecas were created by stacking compost. And so there are these islands that are super nutrient dense. And at the time were used to grow the food that supplied the city of at the time half a million people, which was one of the top five largest cities in the world before the Spanish arrived.  And so I think that there were all these stories that were being told sort of in the front of house context. But as a journalist, I was really interested in, you know, well, what are the stories in the back of house?  And then you have the people working in these kitchens. And I think in Mexico, one of the things that occurred to me as I started thinking about the idea for this book is, you know, it's a country of extraordinary inequality. And it's something that you really feel right away when you move here, when you start living here, because you have people who are living in the city, and then you have a lot of people who commute to the city every day for these jobs in the service industry. And they're coming from a very long way away. The neighborhoods that they're coming from have extraordinary levels of violence, violence against women, disappearances. And it's just these dueling realities of what this city is of people who are kind of living side by side. And I think in a restaurant in the back of house in a fine dining kitchen in particular is a place where people from a lot of different backgrounds are working side by side. And where the hierarchies of a lot of the rest of Mexican life are kind of upended because you might have someone who's from a very poor rural background who ascends to become the chef de cuisine or head chef in the manner of Lalo. I'm going to talk more about in a second. And then you might have someone from a really elite private school who's actually working under them, who's aspiring to be a fine dining chef. And it really is a profession about hard work and talent. It's also about being in the right place. But I think that really intrigued me. What would it be like to put myself in that environment and be able to sort of watch those narratives play out in terms of this place where all of these different people are converging in a relatively concentrated space and kind of map those relationships? And that might be a way of kind of revealing modern Mexico through the lens of a very concentrated environment. So I went to a few different restaurants. I talked to a few different chefs. And I ended up spending a few weeks in one restaurant in the pastry kitchen mostly. And I was interviewing people and I think in a sense I was kind of finding some, you know, the type of story that I thought that I was looking for. But one of the, you know, important pieces of reporting is just logistical. It was a really big restaurant. And it was difficult to observe what was happening in the restaurant as a whole from any one vantage point. And so I felt like it just wasn't really going to work for what I had imagined. And then I was out to dinner with a friend and I was at Maximo Bistro, this small restaurant that was really acclaimed but had a very kind of intimate casual vibe at the time. And my friend was a photographer in Mexico. She just moved there and she was not like a famous photographer or kind of important figure at all. But suddenly the waiter start bringing things to the table that we hadn't ordered. And because we were not the kind of people that are usually treated in that manner, our reaction was essentially like, we didn't order this, you know, don't charge us for this food. And the waiters kept saying, no, the chef, you know, the chef sent this to you. And it was just so interesting to me at the time because I just thought like, who is this person who we're not the kind of people, like there's no sort of transactional benefit to him sending us this food and what is the deal with this guy. And it just intrigued me to know more. I mean, I feel like that story makes me sound a little like I just was excited about some free food that I was getting in a restaurant. But I think for me, it was like, I had never experienced that. And it just made me wonder about who this character was behind the, you know, behind the kitchen and door. And actually at that restaurant, there is no kitchen door. You walk into the restaurant and you see him right away. He's at the pass. He's always there. And so I basically called up the restaurant, I requested to have an interview with him. And we sat down for about an hour and I told him that I had this idea of writing a book about a restaurant kitchen. And he told me this very, you know, just amazing life story in a distilled version where he was born in this town in rural Mexico. He as a young child moved to one of these towns that I mentioned, you know, that's near Mexico City, where a lot of people commute to and from for work. Then he ended up moving to the US with his father, mother and brother when he was around nine years old to work as a migrant farm worker. They went from Florida to Michigan and back and forth for several years. And then when he was a teenager, he moved to the Atlanta area and he worked in a restaurant kitchen. He started working for Chef Eric Repair at one of his restaurants.  He just had this amazing talent. And so he's kind of working at these restaurants. But then on the other hand, he's also this kid who's never really had any friends. He's never stayed in one place for very long. And he starts trying to get in with this crowd of of kids in his neighborhood and who are committing crimes. And he becomes part of this robbery gone wrong of a liquor store and ends up going to prison. And so he's incarcerated for a few years. Then he's deported back to Mexico. His father, it turns out, has stage four stomach cancer. He comes back to take care of him. Then his father passes away. He is caught by ICE. And he has at this point kind of this burgeoning restaurant career in the Atlanta area. He's caught by ICE and he's deported back to Mexico again. And then he has to start over again. And he is now one of the most famous chefs in Mexico, if not the world. His restaurant in 2022 made it to the top 100 restaurants in the world list. But I just, I was really intrigued by him from that story. But I think that one of the things in this profession is it's like you have people who have amazing things that have happened to them. And then you also have people who have amazing level of insight into those experiences. And I think it was extraordinary because I understood from that first meeting that he was both of those things, that he was someone who had had this really epic life story that had touched on so many different issues and contexts. But he was also a really interesting insightful speaker and had a lot to say about those experiences and was really kind of reckoning with this life that kind of fused those two front of house, back of house narratives in a sense. He's someone who can go off and eat at some of these really celebrated restaurants all over the world and he can participate in this fine dining culture. And a lot of the dishes on his menu have big piles of caviar on top and freshly shaved truffles and kind of all of the all of the signs of luxury in that sense. But he's also someone who is always seems to be really struggling with the inequalities that he's witnessing with what's happening to the environment with the way that workers are treated. His father, there's no proof of the reason that he died, but he was a healthy man who died in his mid 50s and the family can't help but believe that the cause was probably all of these pesticides that he was exposed to over many, many decades as an agricultural worker. So it was really interesting to hear kind of from him about anything that was going on in the news that was going on in the world and the kind of perspective that he brought as a person who is in these spaces of luxury and privilege, but who was kind of a critic from the inside. So, you know, I had this idea that I was going to spend like a year immersed in a kitchen and write a book and be much faster than my first attempt. But it ended up taking me essentially the same amount of time to write this book as it did the first book. And I think that I just also appreciated that that long term perspective that I was able to follow him. You know, from these first meetings, like the first scene in the book is someone comes in for a job interview at the restaurant. And that was one of the first moments that I ever witnessed in the restaurant as a reporter. And it just felt to me like a very a very helpful introduction for both me and the reader to this world of kind of seeing the person who's coming from this far away place and seeing the city from their perspective as they go from, you know, one of these communities that most service workers come from into the city, this journey that takes hours, you get there. And he's kind of telling this young woman, you know, this can be a great job, but this is also a really difficult job. And you have to make the choice if you want to commit totally or not. And then I was able to follow him all the way through the pandemic. And to kind of see see kind of an arc, I felt of what he was going through, but then also trace a lot of the backstory of what happened to him over the years and try to provide context. And I think having come from the border and having spent so many years reporting on the border and watching the way the news cycle can kind of treat the border region and package it into these narratives about it being a place of refuge, a place of conflict, good immigrants, bad immigrants. I think one of the things that I wanted to do in this book was just really follow this person's authentic journey without trying to kind of package it around a policy or a news peg or even a preconceived idea of what an immigration story might look like. And so that's something that I, in the same way that because he's a critic from within in this fine dining space that I kind of appreciated about this story and some of the qualities of this story was that he's someone who sort of was always defying my expectations about how I thought those stories might play out. And you could say it's a story about an American dream, but really he doesn't realize his American dream until he's back in Mexico. And so I think that was just one of kind of my takeaways from both of these books and from kind of bringing that border reporting into the interior of Mexico is just that value of kind of widening the lens and what that offers I think in book form that is sometimes hard to do when you're reporting in other more kind of time pressured forms of journalism or storytelling. So yeah, but that's my basic kind of introduction to the book and how I found it. And then I'll turn it over to you, Peter, to see if you have any questions for me. Thank you. Thank you. And anybody in the audience who have questions just a reminder that put them in the slide over and I will moderate them for Laura. So I mean, I was struck in the beginning of the book about it's it's the father is called Lupe. Yeah, I'm struck how, you know, and Lalo, I guess was like nine years old when he was coming as a migrant worker working, you know, picking various kinds of crops in the United States. I mean, it's seen and also that family was split up, right? Because the mother and the daughters were left behind and it seemed like they didn't want, they weren't reunited for many years. So talk to us a little bit about that experience and also from a policy point of view, you know, you've got these nine year old kids who are presumably it's against the law for them to be employed in this way or there must be some sort of loopholes.  Yeah, I mean, I was trying to understand that as well. So just to go back, Lalo, when he was around nine years old, his father came back to Mexico as he did periodically after these long stretches of working in the US and he brought Lalo's mother and brother and Lalo to the US and so Lalo was around nine, his brother was around seven and he had two sisters who were teenagers and who the family decided they would leave in the in the house in the inside of the Mexico where they'd been living and there was an aunt and uncle who lived close by who would kind of keep an eye on them. But his sister was close to getting her her high school degree and she was getting kind of a semi professional certificate as an accountant and they wanted her to stay and finish. And also I think their father felt like this was just a really dangerous life for girls even more so than for boys. And so they they went and they started this journey of, you know, every year going to Florida going to Georgia to Ohio, Michigan, picking tomatoes, oranges, cucumbers, you know, blueberries, mushrooms, onions, pretty much anything that was being harvested along this route. And so it was a really, really, really brutal life. They had come from a very poor background, so kind of the that experience of working in agriculture and working, you know, kind of having these meager living conditions in general wasn't new to them when they were living in the east side of the Mexico they were building their house and they were living in a room with a dirt floor at one point. But there was something much more psychologically taxing about this kind of itinerant lifestyle where you're going from place to place you don't have any family or community support. Some of the employers can be exploitative some not I think in the years my impression he you know he passed away so I didn't get a chance to interview him directly but my impression from talking to other family members is that in his years on his own Lupi had really tried to scout out some of the better employers and spend more time working for them each season. But yeah I you know Lalo was a child when this was happening and so he has kind of his impressions of what the circumstances were but this was one of the issues that I tried to then go to you know people who had worked in the Department of Labor people who had been working for nonprofits to help migrants in these situations to try to get more context of what these conditions were really like what the laws really were from people I spoke to virtually none of what the work that Lalo and his brother were doing was legal. There are certainly loopholes but they're virtually never granted this permission because of concerns about pesticide exposure they were really young. It's interesting now because you know this is one of the things that's happening is that certain states are actually peeling back some of the laws that prohibit child labor. There's a great series this year in the New York Times about you know a lot of young migrants who are coming to the U.S. and who end up working in these illegal conditions inside of factories or farms. So a lot of this is still very much happening. I think that in practice you know part of it is that you go to a place like Michigan and there's no childcare I mean in the summer you're harvesting berries there's no childcare to be found and definitely not that you can afford on the wage that you're being paid to do this work and so kids are in the fields whether they're working or not but it's kind of like there's not much else for them to even do to be in the field all day then kind of help out and because a lot of these crops are paid per piece or per pound you know the more hands you have contributing to filling up a bucket the more money you make so in practice this is still very much going on and while there are you know there are some there is some oversight hypothetically to make sure the children are not working it's very easy to kind of hide kids or pretend they're not working when those inspectors come by so yeah it's it's definitely been interesting to see how you you hear about these brutal conditions that Lolo was experiencing you know 30 years ago and then it turns out that not much has really changed and are they I mean so when these Mexican work agricultural workers come I mean are they are they are getting some sort of temporary visa often for the specific kind of work and their family or how does it work I mean it's it's really depends there there is the H2A visa and that program right now is actually up for some possible improvements but I think one of the things that you know a lot of people come undocumented without papers at all and are working these jobs and then there are visa programs where people are coming and they are enrolled in a program but those programs are almost designed for exploitation because you're recruited from your town the recruiters themselves are very powerful people often in these communities and so if you're recruited to go work for a certain employer say in Florida and you go and you show up and the conditions are really not what you expected there have even been cases over the past year of essentially people who are brought and then are made into indentured servants are working for free to kind of pay back their passage and they end up living in really horrific worker housing with some of them without even a mattress to sleep on without even a place to bathe except for a spigot and a bucket and then they they have very little possibility of logic a complaint or choosing to go work for someone else because they've come to work for this specific employer and if they do complain then that recruiter in their town is not only going to blacklist them from getting one of these jobs again they could blacklist their whole family so the incentives to not complain about these conditions are really high and it's been a very dysfunctional program for many years but that I mean that is that is the way that a lot of people come they do come through that visa I mean if we did if you did the thought experiment where there wasn't any of these temporary agricultural workers what effect would it have on American agriculture and places like the Central Valley and California and elsewhere yeah I mean I think time and time again you know that they've tried to get American workers to do these jobs and people do not want these jobs I mean they are really physically demanding jobs often you know you're in the heat all day long you're working for many hours you're stooped and you're doing kind of a repetitive motion whether it's grabbing the green beans off of a plant or stooping down to dig onions out of the ground over and over and over for many hours there's a there's a good book about this called The American Way of Eating by Tracy McMillan where she kind of embeds with migrant workers and she she does this work herself and you can get a better idea of what that physically is like and then how you're paid in comparison to the costs that you expend along the way in terms of just getting housing or kind of surviving on those wages so they're they're jobs that people just don't don't want for the most part yeah a couple of questions here from the audience um how do the experiences of Lalo and his family shape your perspective on migration cultural identity and the culinary world that's a all-encompassing question yeah I think um you know one of the things that was really interesting to me was his family was migrating um and his mom who's a good cook herself um was also just challenged to be able to to replicate the food that she'd made her whole life in Mexico for her family where's her mother where's the mother from they're they're all from this very small village in the state of Guanajuato Mexico which is where it is in central Mexico it's a little bit northwest of Mexico City it's about a five-hour car ride from where I am in Mexico I believe now and Guanajuato is now one of the most dangerous places in all of Mexico is a lot of organized crime but they're from this very you know I've been there a few times this very beautiful village um that is mostly agricultural and from a very young age I mean she started having kids when she was 15 years old she had her own kitchen and you know her mandate was to take care of her family and cook and so there as they were migrating from place to place she was really challenged to be able to feed them or cook for them food that resembled the food that she'd eaten growing up and so she would have to invent and she would have to kind of try to summon these flavors from other herbs or other vegetables that she was finding along the way um and it was kind of this through line of the book about creativity and migration and the fact that even though today we kind of think of of those innovations as being the territory of a fine dining chef in a kitchen like Maximo Bistro that they've always been there's always been this thread of ingenuity and innovation in the home kitchen in Mexico and just it's kind of part of the migration story I think is this this need to innovate this need to adapt um so that was that was one of the things that I I took away but yeah I'm yeah I'm are you uh cook I do like to cook um I think that this this book both taught me some things about cooking but it also humbled me a little bit in terms of how I think of my own cooking um and then it has coincided the the reporting and writing process has coincided with with my experience of motherhood so um in the second year of reporting this I became a mom and you know there's no diner um who's more tempestuous than a than a small child at your dinner table and who can make you feel worse about your your cooking um my son who's my older child is now a really enthusiastic eater and he will eat anything under the sun and um loves the salad loves escargot the couple of times in his life he's had it well really is a big adventure when it comes to food I think also probably hearing a lot about this book and lalo and meeting lalo has informed his sense that food is this exciting important thing um but yeah I I do like to cook I just think that that is um when you're cooking for kids you're under a clock a lot of the time and it's not it's not nearly as fun as when you one of the takeaways from from your book and also just I mean from what I understand about running a restaurant I mean it's the hours are brutal and very long the work is hard you know it's tough and that's in any restaurant in the world right in mexico city I'm presuming it's yeah with some of the you know you mentioned you know people having to commute four hours to get to these jobs and yeah that woman you mentioned in the introduction when you were talking who was coming for the job interview she basically she didn't take the job because she must have been put off by what it really entailed yeah it's a lot I mean it's you know most most um jobs in mexico are a standard work week is six days and with these restaurant jobs their 12 hour shifts kind of standard and then they can even go beyond that depending on you know what what's happening that night so they're really demanding jobs I think that I just spent a lot of the first year of of reporting this book just kind of marveling at that reality and kind of the the way that people work I think it's also it's also a career where and it's kind of part of what I saw in Lalo's story as he got older and I mean he's still only in his 40s but it's it's not really a sustainable career physically in terms of keeping up that pace and that it is you know he's kind of suffered these different health issues in the last few years and even someone who's such a incredible worker like him finds it difficult to sustain so yeah it it's how has he received the book in his family yeah I think that he like he really likes um one of the things that he's reacted to the most is like the parts that aren't about him the the context kind of about his life and you know I was talking to him and I was observing him but then I would go you know do all this research or interview these experts and I would come back and I would learn that you know round up what he was exposed to is now being linked to this liver disease that he has and I think that there was a lot that that he enjoyed about kind of the the city beyond him there's also a big section in the book about some of the chefs who shaped the the Mexican cuisine that we have today in Mexico City and kind of brought Mexican food to white table costs in Mexico City which was something that I was really surprised to learn that you know in 30 years ago even in Mexico City you really would not find for the most part Mexican food in a fine dining setting and kind of what that says about the enduring biases and inequalities um here culturally and so that that was also kind of a section of the book was trying to understand how that evolution happened and how those chefs sort of set the stage for Maximo Bistro so that one Maximo Bistro opened it was kind of this overnight success that was bringing together French and Mexican food but up until that point um you know that had been kind of a long and difficult road for chefs here I presume uh you know Oaxaca and Chiapas have very different cuisine than say other parts of Mexico I mean there are very strong regional cuisines and how does he draw up those yeah I mean he's a big traveler and so he you know he's gone all over Mexico but I think he's also really interested in he recently went on a long trip to Japan he spends a lot of time in Europe um he's interested in kind of learning from all of these different cultures and traditions and um but yeah there's there's a lot of regional differences in in the food in Mexico I think that's one of the things that depending on who's reading the book you know there are some people who might really already have a good sense of Mexican cuisine but I think that for people who maybe have experienced it mostly through Americanized versions of Mexican cuisine or Tex-Mex to sort of open them up a little bit to all of the variety all of the diversity all the ingenuity of what Mexican food can be and the fact that a lot of what people consume in the country today is really pre-Hispanic um which is so interesting to me because that you know there was such a campaign to extinguish um that culture and campaigns to kind of replace the tortilla with bread that just it's just so intrinsic to the identity of people here that it was it just wasn't possible to to kind of stamp out those practices and in fact you know if you go into the street in the morning and you're on your way to work one of the most common breakfast that you'll find is a tamale with a cup of a tole which are both pre-Hispanic foods that are people are eating as they're you know going about their routine in the morning so I yeah it's been that's been really fun to explore so there's obviously the influence of pre-Hispanic cuisine what about the influence of colonial cuisine I mean yeah yeah I think that that was another piece that was interesting in seeing like the particular that someone pointed that out to me kind of early on like it's interesting that he found this success by combining Mexican food with French food because you know French food is kind of been I think the world over kind of worshiped as the height of sophistication and cuisine and refinement and I think that in a way it makes a lot of sense that in Maximo Bistro you know his cuisine is so lauded because it sort of takes these techniques that we think of as kind of the height of fine dining food or these ingredients like I mentioned before the fresh truffles and the caviar and but then it brings those pre-Hispanic techniques to bear along with them so you know you have you have a mole verde which is combining all of these native herbs and seeds into this green sauce but then it's being served with you know a sea bass that's prepared with butter in the French style you have a mole negro that's served on top of risotto with fresh truffles shaved on top so you have yeah those ideas are kind of combined and I think that's part of where we're making you very hungry Laura so what about sustainable you know how do you sustainably source foods in Mexico City is that something that people are trying to do and are succeeding it's challenging I mean it's funny one of the stories in the book is um this guy Lucio Usobiaga who I interviewed who has this group in the chinampas that we discussed earlier where he's supporting small farmers who are still using these ancient pharma techniques and you know in my house every two weeks we get a box of vegetables from these farmers um and from other farmers in the region who are growing organic produce and it's really evolved over the years but he talks about when he was a graduate student you know 20 years ago he um was working with this guy who said that he was bringing food from the chinampas to these top restaurants in Mexico City and in reality he was going to the central de abastos which is the big just market public market and buying stuff and putting it in boxes and pretending that it was from the chinampas and I said how could you tell that that he was lying and he said well he was bringing pineapples that were supposedly from the chinampas like you can't grow pineapples in Mexico City you know it's not the right climate so it is this very obvious fraud um but there just wasn't the kind of integrity to sourcing ingredients um even in these top restaurants to really understanding where things were coming from and how they were being grown I think that's that's one of the biggest influences that Lalo has had on the dining scene here because he's someone who you know it's not enough to him to just say this comes from this place he worked on farms he really understands the process of farming and growing food and not only from the industrial fields where he was you know a migrant farm worker but from his village where the town where he comes from where his family worked in agriculture um and so he uh you know he'd go to that to that farm he would go to that coffee plantation he would go to that um you know that area on the coast where they're diving for scallops or to see where the cows were being raised for the beef or so he he's really immersed in kind of the the sustainability and the the growers and understanding where those products are coming from and so he's kind of become this person who has created all of these pathways to sourcing with integrity from these producers because he's kind of identified who those producers are and he's made them known you know to this larger restaurant community that then kind of follows and starts getting their chickens from this purveyor or starts getting their you know cheese from this purveyor um because they they trust his judgment and so um he really strives for most of the ingredients that are served at the restaurant to come from Mexico although there are exceptions um but it's it's become a really that's become a big part of the ethos of the restaurant and I think that that really comes from those experiences of seeing the contrast between you know the mill bus style of agriculture where he grew up which is a system where um the plants are intercropped where you have the corn that's serving as a trellis for the beans you have squash that's um protecting the ground from weeds and insulating moisture with its leaves um and also a lot of other vegetables are kind of intercropped with those three sisters that we we always talk about in the US um and seeing the contrast between that and all of the milpa all that the milpa kind of provides um in the symbiotic relationship and the monoculture and the industrialized fields and the tomatoes that they harvest before they're even ripe and the cucumbers that just taste like water um because they're so massive and really appreciating not only as a consumer but also as a chef you know the difference in flavor and quality between those products so I think that his appreciation for sustainability runs much deeper it's not just about like a catchphrase or an idea um a pretty idea I think it's really about like this kind of creative integrity that goes very deep um in his restaurant but yeah it's challenging and it's like it's hard because it's a it's a movement and it is making a difference but you also don't want to sort of exaggerate the impact that it's had you know Mexico is a country that's going through profound environmental destruction um from the trend maya that is being completed and is about to open that was built over these cenotes and the yucatan that are really fragile ecosystems um that most environments will tell you it's not a good idea to build a train on top of them um or to raise the jungle in the name of of building them a turn away from sustainable and energy wind energy towards more oil um so you have these kind of contradictions at play in Mexico between the narrative of sustainability in a place like maximo bistro and the realities of what is happening to the environment in a lot of the country um but I I've come to think of these chefs as almost people who are conjuring a kind of utopia for what they hope the country could look like and they're sort of that is part of the narrative of what they're telling is they're trying to communicate you know the value of these products and the value of these practices um before they're lost another question from the audience um how do you approach telling an immigrant's life without romanticizing it yeah I think it was something really important to me um in part because of what I was talking about earlier with having lived on the border and having seen you know the way that the narrative seems to flip-flop back and forth between you know these empathic stories about people's struggles to come to the US and these stories that are describing you know this is kind of an invasion of people um to the US and I think that one of the things that was exciting to me about Lalo is that he's this person who really um speaks with this candid kind of reflection that um sometimes it seems like he's really not speaking as the person that you you expect in terms of what you imagine a question would solicit to a reporter um he doesn't tie bows around things he's very honest um and so for me part of it was just really trying to capture him as a person and kind of the full scope of him as a person on the page and having it not be reduced to um the usual beats that were accustomed to in these stories so one of the examples that I think about is in these stories the the crossing of the border is usually a key plot point in a narrative about a migrant you know that journey the uh which which these are very dramatic journeys I mean there are record numbers of people right now crossing through the Darien Gap and they're on their way to the US and these are incredibly dramatic journeys but in Lalo's experience those journeys were pretty easy he crossed a lot of the time in a car he was a young child at one point he crossed through the fence but he took a bus to get to the I mean it he was with his family um they just weren't those kinds of dramatic journeys and when we would talk about his life and we would talk about his stories he didn't seem to dwell on the boarding border crossing stories um a lot and so I think I kind of followed that lead in the book you know the the border crossing moments in the book are not big chunks of the book they're not whole chapters they're moments in time um and I think that that kind of subverts in a way the idea of what a immigrant story is supposed to look like um but yeah I just I really I think he's he's an incredibly charismatic person and a very complicated person and I think I just really tried to do justice to that um in the book and make him a three-dimensional and just convey that and I think the other thing I would say is that you know this book's called The Migrant Chef and part of that title is about sorry someone's revving their engine up my window I don't know if you can hear that but um part of that title is about migration but part of that title is about him being a chef and so a lot of this story is about that journey of someone who's just kind of um on a creative path and finding themselves creatively and to me um that was kind of as much of a through line or as much of an arc of the story as the migration piece um and looking at him through that lens of just I would want you to be interested in this guy regardless of the issue that he's tied to I want you to just be compelled by him as a person and kind of his personal journey the book is The Migrant Chef thank you very much Laura for the brilliant book and please buy it and thanks for listening to this and thank you Laura for the presentation thank you so much thanks Peter


Hi, everyone. Happy Saturday, Joe for Jasper's casebreaks.com. That's me and that is Jason Tatum That's who we're looking for in this 2017 2018 Panini Donnery's optic blaster basketball break ten blaster boxes half of a fresh case Break number one and one spot gets you two teams Right here. Are they really going for that much? I mean with the yeah, probably a good idea to get in before the Celtics go deep into playoffs and if this guy plays well Big thanks all of these folks for getting into the action. Let's double your names up. Oh double up Oh, oh, and they're all teams are in right there, too So let's roll it. Let's randomize it six times one and a five for each list one two three four five and six and final time after six times we've got Rockyo down to Matthew and then six times for the teams one two three four Five and sixth and final time after six. We've got the San Antonio spurs down to the Nalins pelicans Unfortunately the pelicans not as desirable in 2017-18 But you know be some surprises in there All right, Rockyo with the spurs Sean with the Nets trailblazers from Matthew Tyler with the T Wolves Matthew G with the Warriors Jeremy with the Suns Matthew S with the Jazz other Matthew S with my Lakers Matthew Silverstein Matthew Shira with the Wizards Oliver you have the Rockets guy with the Pacers Sean Maddock with the Pistons and Magic Stephen P with the Raptors Tyler with the Hornets Rick with the Cavs Matthew Silverstein with the heat Rockyo with the Hawks Rick Serrano with your other last-bought mojo team Clippers We've got Cavs Clippers Stephen with the Mavs Sean with the Bulls Matthew Shira with the Nicks Matthew Garner with the Sixers Matthew Silverstein with the Memphis Grizzlies Jeremy Agnew with the Kings there you go Matthew Shira with the Boston Celtics my rivals guy with the Bucks Matthew with the Nuggets Oliver with the Thunder and Matthew with the pelicans Right. Yeah in 2017-18. Yes Zion probably drinking Capri Suns At this point after high school Not first high school days, let's sort by column B now I'm gonna switch trades are allowed, but I'm gonna switch away from the screen just for a second because I'm gonna pop open that fresh box And we're gonna see We're gonna see which ten boxes we're gonna do She's gonna need a knife pen at this rate too. All right. All right, so let's pop this guy open Stephen pieces Capri Suns are delicious. I didn't say they weren't delicious. I still drink them I Still have I still have trouble putting the straw in the pouch though That's that's that's an extra level of dexterity that I do not have It's kind of all over the place here. Joey. What's going on? We got T. Wilden Hornets for trade All right, so you can see on the top camera. Here's two four six eight ten boxes right there And then two four six eight ten Right over here.   I'll leave those there I'm gonna let's bring it back to the list And I'm gonna pause the video and allow for a little bit of trading time and when we come back We'll see if there's any trades and then the break stick around All right, welcome back folks a little bit of trade chatter But in the end no deals were done. It's all good though. Thanks for hanging out with me on a Saturday ladies gentlemen Thanks for spending a bit of your Saturday with us Break with us watch baseball with us And we'll have a nice whole Saturday night. All right There it is. So let's Let's open up a few boxes here at a time Maybe five boxes at a time Now basketball starting up soon too, and that's exciting baseballs underway basketball maybe they're scrimmaging Getting some poops action going soon. I think there there are some teams that are gonna Basketball teams are gonna benefit from the the long layoff, right? Trailblazers got healthy Youssef Nurkic got healthy. I think a Collins. Which Collins do they have? Do they have the Zach Collins or the John Collins? They have a Collins on that team. I think he's he's effective eat up some minutes He's healthy Denver Nuggets bowl bowl. I Think was battling foot injuries for for a lot of the season and then he has Gotten a little healthy, and I think maybe a player or two were Offing out or sick or something. I don't know what they're but I think he's got opportunities now Oh, they have Zach Collins this way So Bowl Bowl is now getting some opportunities because he's rested up. He's healthy Right one scrimmage and Bowl Bowl cards through the roof Imagine nuggets are a playoff team, right? If nuggets do well in the playoffs and if Bowl Bowl is getting some decent minutes a couple highlight plays here and there That's that'll all that's that's all it'll take You know for that height Thanks, even P. Yeah, that the Rockets out. Let's flip back to I have a Rockets pick I've got some hoops picks futures pick. I rarely do futures. I'm not very good at those but But for the eighth seed In the western conference, which means possibly someone winning a winning a play-in game as well for the eighth seed I go I went Portland plus 275 Pelicans plus 325 just for Zion reason hashtag good for the hobby and the Spurs plus 1800 Coming out of the out of a long life and a pandemic who benefits the most probably good coaching Teamed with depth and good coaching I think Spurs Spurs are there eastern conference champion I got I'm only taking long shots really. Yeah, that has the most value I mean, it's probably going to be the bucks but or the Raptors But plus 600 for the Celtics and I picked these ages ago weeks ago So those odds might be different but six to one for the Celtics Why not another well coached team Jason Tatum? That's not Jason, but Jason Tatum is in this set You know, he's playing well Miami another well coached team D team jimmy butler a couple young rookies Kendrick Nunn either hero can make an impact So I like that as well Miami 18 to 1 plus 1800 Could be great western conference champion pick Dallas plus 2020 to 1 Hey, if if luka donjic and christos porzingis just go off That'll be great and the picks even p was talking about was my houston pick 10 to 1 plus plus a thousand For the houston rockets to win the west now most likely it's going to be my Lakers or the clippers right but if i'm investing a little If i'm investing financially in the futures the futures market Right like oil like for a team. I think there's some value there well rested harden Right so they can just ride hard and 40 minutes a night Have them shoot like a million shots a game I'm gonna do that west rook is there as well probably the best player that Harden has played next to since the west rook Durante's maybe or just harden durante's It's been a while since he's had a had a good second player chris paul maybe But yeah Landon's asking do you think the pelicans can Could have gone to the playoffs with if they had zion the whole year and healthy and no corona. Yeah, I think so I mean small sample size, but zion has looked pretty incredible You know in the handful of games the dozen or so games that you play there's anis rodman hall kings but I It's crazy to think lily if To think that zion value could go even higher, but I spam out of bio what happens if zion gets the playoffs Right That's that's another boost in value right Let's say he meets the lakers They're the ace he meets the lakers and has a great series against the lakers Lakers very public team everyone's going to be watching them on on tv That's another boost in value would imagine What if zion is the reason they beat the lakers? Imagine that So yeah, I mean, I think they're still still probably could still be some room For for zion to go up even more I mean, I'm sure those playoff prices will will end up You know kind of dipping and plateauing back down again, but why not? This yannis will go to guy and the bucks There's poor zingus nix addition and our first jason tatum Rated rookie base and we were looking for hollows. I think those are our main Chase matthew shear and the cell takes But this base version is not too bad as well Ricky rubio for the jazz so those blue borders not numbered, but they will ship I think it's only veteran commons that that aren't going to ship Obviously rated rookies and hollows and stuff will all ship Battleship inserts will ship buddy healed. That's a parallel that'll ship Isaiah Thomas court king's lime to 149 will ship Wow, are those are those jason tatum rated rookie base rich is saying 80 bucks raw There's john collins Hawks zack collins on the trailblazers Those tatum prices have been steadily steadily growing for for months now and if he has a good playoffs Could be could be even better More yannis will save more of those yannises All right, buddy healed in the back Another five boxes to go Let's rip all those open Why why why dwayne bacon lol? What's with the lol tyler? Dwayne bacon before the the pandemic put the season on pause in a glee game. He had dropped like 40 points I'll bet if the season kept going that bacon That bacon would have made this made the charlotte hornets played really well And all those people who got all those dwayne bacon cards back and when we were ripping all this 2017-18 stuff So before you laugh, but i'm pretty sure look it up tyler. In fact, check me But i'm pretty certain that pre pandemic There was like a glee game where dwayne bacon had had dropped 40 in atlantic glee game But he had dropped 40 or something like that and I was like and at the time I was I remember being like Dust off those dwayne bacon cards Did the celtics do better with or without kairi or they seem to do okay this season without kairi Yeah, save your bacons tyler. Don't don't waste your bacons A wasted bacon Can't waste bacon Wow rich is saying you we may not get is that something that you heard from psa or becket rich I guess they are pretty backed up on grading. In fact, I think was it jack b someone was saying I think it was basmati and he was he was saying that he got an email from psa. Maybe it was you rich Got an email from psa saying that they were hiring I guess jackson southern california. So maybe that's why he got that email, but I think it was him, but he got uh That they were hiring people graders Yeah, I think that celtics eastern conference play is not too shabby at six to one Tyler like And I got that weeks ago. So that those numbers might be different today But um, I mean most likely it's going to be bucks Right most likely or or second most likely probably toronto, right? But uh, but if it weren't those teams, that's kind of where my head's at with futures bets I don't really play a lot of futures And then I'm I'm we're talking like maybe lunch money on each of these things, but So it's I don't go too crazy on on on any of this it's for entertainment purposes anyway, but But I always try to go for that sort of longer shot value Where's the value with the longer shots rex? What's going on? I know I've played against your cubs today And I was rewarded Nick Stover, what's going on And even longer shot at miami eastern conference plus 1800 18 to 1 That's not too bad, right I'm talking about Jimmy Butler goes off if Kendrick Nunn and Tyler hero or I had a schedule They still have bam on a bio on that team. He's playing some good basketball. Spolster is a good coach Uh Google man, do you know when the 2020 luminance break will be? No Whenever it sells out though, probably tonight I don't know. I know we have to do something about that passing the torch I feel like I have to maybe after I got a couple more breaks to do but after a couple more breaks Maybe I'll show you the video of the break that nick jaspy did of passing the torch football and show you what's inside I think that passing the torch was an online only release. So maybe not too many People who have experiences with that. So maybe they're a little shy all right Last five boxes second half is in the store jaspy's case breaks.com and ivenrab Is your rated rookie autograph for the gris matt silverscene with that one? Yeah, let's get that mamba mixer going too. I think there's a few mini breaks that we need to accomplish But I think we should be able to knock that out tonight as well Donovan Mitchell rated rookie nice Dodgers are down five one to the giants It's only the fourth inning bottom of the fourth dodgers can put up a crooked number pretty easily Oh, they got virtual fans in the stands. That's weird. I think that might be a fox broadcast It's interesting justice windslow zack collins rated rookie See on the back Dodger state. I know about other stadiums folks a dodger stadium is allowing Pet photos as cardboard cutouts that you can get back at the end of the season Authenticated by the mlb. You'll get a little hologram sticker on it might have to do one of those I saved some of these lebron james's as well cabs rick I know he even his His cards have been gaining some value on the secondary market obviously because it's Obviously because it's lebron james Then people are kind of that those those base cards like this. Yannis too Over the last couple years. I've just been really increasing in value I'm not sure Uh, actually, no, brad. I think I read the rules and it said one pet per photo And for the human cutouts, I think it's just one One human per photo although I think I saw some people sneak in a A second human a baby sized human Into their shot. What about steve curry hollows? What's the situation on that secondary market? Malik monk rookie kings to 149 al-horfer devon reid Hall kings mitch richman and joe johnson for the jazz at the very end. There you go ladies and gentlemen nice quick break Of 2017 2018 panini donner's optic basketball 10 box blaster edition random team break number one second half in the store on jasby's case breaks Dot com i'm joe and i'll see you next time. Bye. Bye


We're not going to do much new stuff, so of course last time we talked about the first principle of finite induction, I didn't do any problems. I was trying to squeeze everything in because we missed, you know, Tuesday.  But we'll talk about that, and basically the only thing new that I'm going to do is talk about that principle, and then we're just going to try to get through a few problems, try to, you know, get you prepared for the homework. Induction, of course, as I said before, this is a very, you know, I mean, this is a very common proof technique, not just in number three, but in mathematics in general. Most of you have done this before, but I just want to, you know, kind of go slowly through the first part just to make sure that everybody's sort of up to speed on this stuff. And then we'll probably pick up the pace a little bit, but initially, you know, I think it's best to err on the side of caution and go more slowly than more quickly in the beginning. So that's the plan.  Then I'm going to do a problem using induction, and then like I said, we'll move on to the second principle. So what I'm going to do, for the most part, I'm just going to start the labeling over again. But this is just a continuation of what we did last Thursday. Okay, so again, let's just, hopefully this is going to get better here soon. Let's recall theorem one, which was the first principle, like I said. And again, this is called just the first principle of, I'm just going to call it induction. I don't have anything to practice on, unfortunately. This doesn't belong, as far as I know, this doesn't belong to the math department, so we don't have any of these gadgets. And I don't have an iPad or any of these things, so I just, I don't have any way of practicing, unfortunately. Okay, I'm thinking about writing in cursive, and I think that might make this a little bit easier. Okay, so we've got our set S that's contained in N. We talked about this before. This N denotes the set of positive integers. Okay, so the first condition was that one is an element of S. Of course, if you have this in your notes, there's not necessarily any good reason to write this down again, unless you just kind of want to reinforce this and you think it'll help you. But the second condition is that natural numbers N, or in other words, for all little N and capital N, if, and this is a conditional statement, if N is in S, then the next positive integer is in S. And the conclusion is that this subset S is actually equal to the set of all natural numbers. And I try to give you kind of a heuristic argument as to why that was true. I said I would give you a formal proof using the well-ordering principle, but because of time, I don't think I'm going to do that. I think time would be better spent just talking about some problems and kind of showing you what I'm expecting in terms of work. So, of course, we're going to get to plenty of the formal stuff here when we start actually, you know, proving some of these exercises. So I don't think I'm going to say anything else about this for now. So what I'm going to do is, let's see if I can, hopefully I can squeeze this in here. I'm going to start, again, I'm going to sort of start slowly here. And again, those of you that are old hands at this, I apologize if you're bored, but I just want to make sure that everybody is on the same page here. So let's do an example. And so let's look at the following. The book, this is actually a little bit easier than the example, I think the examples that the book starts with, but I want to start with something that's easy to digest. Okay, so for every positive integer n, let's let, sorry, let's let p sub n be the statement. I'm going to put this in quotes. Let's see, I'll tell you what, I'll tell you what, sorry. Let me, let me change the notation slightly here. Oops, wrong thing. Okay, p of n, instead of sub n, okay. And it's going to be a statement. And it's going to be 1 plus 2 plus 3 all the way down to n equals n times n plus 1 over 2. Is this homework? No, it's okay. Is this actually one of your homework problems? Okay, well, that's fine. I don't care. Let's see. I thought, I thought why they said we're a little more complicated than this, but okay, well, that's, that's fine. That's okay. I'll, I'll still talk about it. It's okay. Okay, okay. Okay, well, that's, okay, good. That's even better, actually. So, yes, I'm not used to this. I'm not used to this. So, yeah, I'm going to try not to treat you like your Calc 1 students, but this is all I've been teaching since I've been here. So, I'm very glad about that. Okay, so I want to be clear about this too. This is a, this is a proposition. Okay, we're going to end up proving this, that this is actually true for every positive integer n, but for now you just treat this as an assertion, which may be true or false. Okay, it is true, but in general I don't want you to think of it this way. This is just a statement. Okay, so just to make sure that everybody is on board here, the first thing I will say is, let's just actually look at a specific case of this. P of 4 is, what, it's, okay. So, this statement just says that this, the assertion is that the sum of the first n positive integers is equal to the very last one in the sum times the next guy divided by 2. That's what this is saying. But don't get hung up on the letter n. It could be k or it could be n plus 5. It's just, the way that you interpret it is 1 plus, whatever that last guy is, that sum is that guy times the next guy over 2. That's what it's saying. Okay, so P of 4 then of course is just the statement, and it's 4, right? So this is just 1 plus 2 plus 3 plus 4 equals 4 times 5 divided by 2, right? That's what P of 4 is. And you can check this. Alright, so the left side is 10. The right side is 10. So this assertion is true. So, I was going to do some other ones here, but of course, I assume you guys are, you know, you've done this stuff before. You know, if I said what's P of 3, you could tell me exactly what that is. So, what is, what's P of 1? I'm not going to write this down. What's P of 1? What's that assertion? It's an equation, right? It's not a number, right? So just, yes, that's right. I'll just say it, there's sort of a shortened version. P of 1 then is 1 equals 1 times 2 over 2, right? Because n is 1 in this case. You can write that down if you want to. I'm not, this thing is already still awkward. I don't think I'm going to do that. But okay, so hopefully that's clear, at least what these statements are. Okay, and what we're going to do now is actually prove that this assertion, this P of n assertion that I gave you is actually true for every positive integer using this first principle of induction. Okay. So, let's call this example 2. Now this, I'm going to, some of you may ask this, like, oh, do you expect me to write all this exactly as you write it in the homework? No, I don't necessarily.  Before the end of class today, I will try to tell you a little bit more of what I'm expecting as far as streamlining some of the things that I'm saying. But for now, I'm just trying to get the idea across as clearly as I possibly can. So I'm going to say a little more than maybe you might say in your homework. Okay, so the problem is to prove that this holds, as I said, for every positive integer.  Okay, so as I said, I'm going to take my time on this. Okay, so I already gave you example 1, so this p of n that you're going to see is just going to be the same p of n that I just gave you on the previous page. Okay, so we must prove that p of n is true, all natural numbers n. I'm going to switch back and forth between using natural numbers and positive integers just because I want you to equate these in your mind because they are the same thing. Okay, so going back to this first principle of finite induction, if you remember, if you have a subset of n that has the property that 1 is in there, and whenever something is in there, then the next guy is in there, then that subset has to be all of n. And so if we want to prove that this statement is true for every natural number n, I'm not going to write all of this out, and you don't have to either. But the point is, here's the idea. You're going to let, even though we're not writing this down, we're going to let s be the set of positive integers and for which p of n is true. And we want to show that s is everything. In other words, p of n is true for everything. So using this first principle of finite induction, it's suffice to show that 1 is in s, and whenever n is in s, then n plus 1 is in s. In other words, p of 1 is true, and whenever p of n is true, then p of n plus 1 is true. And then the first principle of induction then gives us that s, in fact, has to be everything, which is exactly what it is we're trying to prove. Okay? So let me just say what it is exactly that we're going to do. So we will show, I'll label these here. So 1, p of 1 is true, and 2, some of you that have, those of you that have taken to the screen may recognize this first, this first task as the inductive step, or sorry, the base case of the induction, and the second, the inductive step. You may have heard that terminology before. Have you heard this? Okay. Whenever p of n is true, so is p of n plus 1. Okay, so then if we can do this, then theorem 1 says that p of n is true for all n. Right? So let's just go ahead and do this. Let's go ahead and verify these two conditions. So the first one, I'll just circle this. I should say, and again, those of you that have done point of induction know this, the first part showing that your statement is true for n equals 1 should generally be very simple. There shouldn't be something you need to do much for. And in general, some of these problems you're just going to say, duh, obvious. There's nothing to say. If p of 1 is 1 equals 1, well, okay, that's true. You don't have to go into a long explanation. Well, by the tautological philosophical principle number 4, don't do that. Things that you know from second grade, you can just assert. That's fine. So, as I said, so the first thing we're going to do is just verify that p of 1 is true. And as we said before, I'll actually write this down. p of 1 is the assertion that 1 equals 1 times 2 over 2. Right? Okay, so everybody clear on this point? And all you have to say is clearly true. That's it, okay? 1 times 2 over 2 is 2 over 2, which is 1. You don't have to write all that out. Just clearly this is true. Okay, so now the second part, I'll tell you what, let me just wait until everyone's copied this. I think I'm going to go to a new page so I don't get stuck in the middle. Okay, is everybody okay? You want to have this down now? Okay, all right. So the second part is you'll just say, we're going to assume that p of n is true and we must show that p of n plus 1 is true. Something you might find useful as you're writing proofs is to just keep track of what it is that you're assuming and what it is that you want to prove. And it can't hurt just to be very explicit and say it, not only for my benefit but for your benefit so you know exactly what it is that you're trying to do. So again, remember this principle of the second condition, from theorem one, part of the hypothesis is that if p of n is true or if n is an s, then p of n plus 1 is true. So it's a conditional statement. If then, this is an implication. So what you're doing is, if you want to prove an implication, you might say, okay, well if it rains tomorrow, I'm going to stay inside. So you're not really, in some sense, you're not really saying anything about what's going to happen if it doesn't rain. You're saying, but if it actually does happen to rain, then you have a conclusion. So an if-then statement, if you want to prove it directly, what you do is you assume what's called the antecedent or the hypothesis, and then you prove the consequence. That's the direct way of doing this. And so in general with induction, that's what you're going to do. So I'm going to just write it this way, and I'm going to use this language because I think it's a little bit suggestive, which is what I want.  You can use this fact, what I'm about to write, you can use to prove that p of n plus one is true. Okay, so we're assuming p of n is true, so that means that, of course, in example one, I defined what this meant, right? So there's no, nothing's really left to the imagination here. One plus two plus three, all the way down to n, that sum is n times n plus one over two, right? That's our assumption. And we have to show, and I'll label this a little bit differently here, let me use an actual star here. Okay, p of n plus one. So we talked about what p of four was, right? Here's p of n. So p of n plus one then is just, instead of the sum from one to n, it's the sum from one to n plus one, right? On the left side. And instead of n times n plus one over two, it's n plus one times n plus one plus one over two, on the right side. Does that make sense? Okay, so that's what it is that we have to show. Okay, so I'm going to condense this slightly. So this just becomes n plus one times n plus two all over two. Okay, so everybody see where I'm getting that? Yeah, okay, good. Okay, so here's what you're going to do. And in general, you may have, there are some problems I know. I think there's one in the homework that deals with an inequality called the Bernoulli inequality. So you may not always be proving the qualities. You may be doing inequalities. So the point is the assumption, this p of n assumption, as I said, that's why I'm using the word have. You can use this. You can do whatever you want to it. For all intents and purposes, it is a fact. It's true. And you use that to get to where you want to go, which is here, okay? So if we know that the sum for one to n is n times n plus one over two, we have to prove that when we go all the way up to n plus one, we get n plus one times n plus two over two. And this is what we have to work with right here. So think about what we need to get on the left-hand side. We need to get one plus two, I left this out, but the term before the n plus one is n, right? Okay, and we have that up here. So what's the only difference between the left sides of these equations? Well, the bottom one has an n plus one, whereas the top one does not. That's the only difference. And I can use this as fact, the top one. So if I want to get down to this, it seemed reasonable then that we might want to try to add n plus one to both sides. Because then at least the left-hand side is exactly what we want. And we just hope that the right-hand side can be made into what we want by factoring or something. That's the idea. You guys with me on this? Okay? All right. So that's exactly, as I said, that's exactly what we're going to do. So we're just going to add n plus one to both sides of this asterisk equation. Okay, so when we do that, what do we get? Well, we get n times n plus one over two plus n plus one, right? Anybody with me on this? Okay, so what is it that we want the left-hand side to be? We want it to be n plus one times n plus two over two. That's what we want to prove. So all I have to do is try to make this into that. Okay? You're just going to fiddle with this using some basic algebra and it'll fall out without a whole lot of work. Okay, so let's see. Well, there are actually multiple ways you could do this. I'm just going to, I think, go with the most natural approach here. Okay, so we want this to become n plus one times n plus two over two. So one thing that might be a reasonable thing to do first off is to get a common denominator here, right? Because then at least we can combine and we'll have a two as our denominator, which is exactly what we want, and we just hope the top will factor. That we're done. That's it. Okay, so n times n plus one over two plus two times n plus one over two. Okay, and as I said, I'll just go through this kind of slowly, just make sure that everyone's with me on this. So we can combine the numerators here, of course. You're not basic algebra students, so I assume you guys can follow this. And so, even so, I'm still going to go through this kind of slowly. So this is n squared plus n plus two n plus two, right? Which, of course, we're going to combine even more, but that's certainly true. Okay? And so what is this? This is n squared plus three n plus two all over two. Now, what we want, right, is to get the right hand side to be n plus one times n plus two over two, so what we want is for this to factor is n plus one times n plus two, and it does, the numerator, right? And that's it. You're done. Okay. Now, really, at this point, you might say, well, what do I need to say at this point? Therefore, you know, by the first principle of finite induction, this p of n is true for all positive integers n, sure, that's a nice way to end it. If you want to just end it like this, that's fine too. You've clearly established what you need to do establish, and any, you know, well, of course, mathematicians don't prove things as simple, really, but would say, oh, well, yeah, of course, that's done. You've done exactly what you wanted to do, so you're good. Yes? No, I don't. I mean, there's a lot of different conventions here.  I mean, there's lots of things that you can say. Oh, I'm done. LOL, yeah.  Don't say that. That's disrespectful, but, no, you can do whatever you want. I mean, if you have a convention that you like, that's fine. But you do want to end it. Oh, I mean, if you wanted to leave, to finish like this, that's okay. You have done what you needed to do. And that's fine. Yes? I did things a little differently. So I would take the star equation and copy it out there on the left-hand side of it. I would replace the 1 plus 2 block up to n with the right-hand side of the asterisk equation. Right, right, right. And then on the total side of the equation. Yeah, I know what you're saying, except the problem is that you don't want to make assertions being true before you've actually proven them. So the slight issue is saying n times n plus 1 over 2 plus n plus 1 equals n plus 1 times n plus 2 over 2. That's what you're trying to end up proving eventually. So I really would rather you not do that because you are making an assertion that hasn't been verified yet. So the point is don't make claims until they've actually been verified. Here, we haven't done that. We haven't assumed what we're trying to prove yet. I know what you're saying. Then you just simplify so that you get 5 equals 5 or something like that at the end. But that really is not the best way to write the proof because there's a part where you're assuming what you want to prove. Hang on one second. Therefore, I can replace anything that I see. No, but you haven't proven this yet. But if your next step is to write, and maybe I miss understanding what you're saying. I know what you're saying too. We were taught to do it the same way. Okay, do you have it written down maybe? Yeah, and he's plugging in the quality for 1 plus 2. Oh, I see what you're saying. So you're just saying at this point right here. You're saying here, then just replace this with n times n. Yeah, yeah, yeah, that's fine. I thought you were saying something else. And that's what I thought you were saying. No, no, no, sorry. What I thought you were saying is something that a lot of people do. And so that's why I kind of interpreted it that way right away. But yes, that's fine. That's okay. Yes? I think I saw what you thought he was saying. Oh. If that was the only way you could see to do that, wouldn't you have to assume it wasn't true to derive a contradiction? That, I mean that, okay. It wouldn't be pretty. Yes, yes. Logically that would be okay. Although yes, like you said, it would not be a pretty way to go about it. But that's one way you could circumvent that issue. But yeah, what you said is fine. And just to be clear, it is not the case, certainly with proofs, that there's only one correct proof in general. That is not, that's certainly not the case. The only thing I want to, want you not to do is what I said before is assuming what it is you're trying to prove somewhere in your argument. That's the only thing I want you to avoid. Is it too much true therefore it is? Yes. No, no. I mean, you know, really I'm looking for, for these induction proofs, using the first principle of induction, what I'm really looking for is that you, clearly established the base case and the inductive step. As far as actually mentioning it explicitly, you don't really need to do that because, you know, this algorithm is very well known. And so you don't, it's not, of course it's not wrong if you're very clear and verbose about things, but it's not something I expect that you would have to do. Any other questions? Yes. For proof by induction, or is it just like, use whatever method you want? So, that's a good question. Because I haven't come close to writing the exam yet, I don't know, honestly. We can definitely talk about that closer to the time of the exam. I mean, you know, if it has any problem, use induction, we don't need that. Yes, yes. Are you using the principle? No, no, no, no. But yeah, but, you know, also if I say, you know, use the first principle of mathematical or finite induction to prove blah, blah, blah, then if it's specifically, I say that, then yeah, I would definitely expect that you would do that, for sure. But no, you wouldn't have to be that explicit. Any other questions about this? No? Okay. Good. All right. So, what we're going to do now is, and this is, like I said, this is the only new, this is the only new thing that we're going to talk about today, is the second principle of finite induction, which by the way, I won't get ahead of myself. Let me just go ahead and write this down first. So, second principle of finite induction. By the way, this, everybody else in the universe calls this strong induction, okay? So, just letting you know, in case you see this in a future math course, strong induction, that's what this is. I'm going to just stay consistent with the book unless the book says something that's actually wrong. So, I'm just going to call it this, because that's what it's going to be called in the homework and such. But yeah, that's what everybody else calls it. Okay. And it just says this. Suppose that S is a subset of N, and it satisfies two conditions. First condition is that one is an element of S, this is just like before. And the second condition is a little bit different for all natural numbers N. If, okay, so, here's the difference. If all the numbers, all the positive integers, less than or equal to N, are in S, then N plus one is in S. And the conclusion then is the same. Then the set S is actually equal to the set of natural numbers. Okay. So, I'm probably not, just for the sake of time here, I'm not going to go into a written explanation of this. But you can see the difference is that for the second condition, the hypothesis is stronger now. Instead of just N being in S, if you remember that was the first principle of finite induction, now everything less than or equal to N is in S. So, why is this true? Well, it's a very similar argument as before. So, why does S have to be N equal to N informally? Well, the first condition says that one is in S, so the smallest natural number is in there. The second condition says that whenever you know that one through a certain number, all of those are in S, then the next thing is in S. So, the second condition says, because we know one is in S, two has to be in S. Now we know that one and two are in S. And by the second condition, three has to be in S. Now we know that one, two, and three are in S. So, by the second condition, four is in S. So, that's the idea. As I said, I want to spend more time actually working problems than going over the kind of the foundation. So, intuitively I think it should be, if you think about it, it should be pretty clear that this is what's going on. So, what's the point? Why do we have these two principles? Well, here's the idea, and this is where the word strong comes from. Very roughly, and this is very informal now, if you can prove something by the first principle, you can prove it by the second. Okay. Why is that? Well, okay, this is kind of a mouthful, but I'm just going to say this. So, if you can prove something by the first principle, what is it that we, what was conditioned to for the first principle? If N is in S, then N plus one is in S. So, basically, if you can prove that N plus one is in S just by knowing N is in S, then you can certainly prove that N plus one is in S knowing that one, two, three, four all the way up to N, or in S. It's a stronger assumption. Okay, so that's the idea. And so now there are situations, and I'm going to do an example for you here in a second, where instead of just, so to prove that P of N plus one is true, it's not enough just to know that P of N is true. You actually need to know that the other guys below N plus one are true as well. And so, again, I'm getting ahead of myself now, but I'm going to say this and I'll repeat myself again, but how do you know the difference? How do you know when to use what? And it's basically just when you have a situation like that. When to prove that P of N plus one is true, you can't do it just from N. You need maybe N and N minus one and N minus two. That's when strong induction, or the second principle of finding induction applies. Okay, so let me, just because, again, I want to get through a couple more examples here. Let me do this example. Actually, what I'm going to try to end up doing is number 13 from your homework. You have to do number 14. I'm going to talk about number 13, which uses strong induction. I will say strong induction because it's just easier to say, but that's what, I mean the second principle, right? Okay, so suppose that the numbers A sub N are defined as follows. And so, again, number 14 is similar to this. Okay, so the first three are just explicitly specified. A sub one is one. A sub two is two. A sub three is three. Okay, and after that we have a recursive definition. A sub N is the sum of the three preceding guys.  So then, okay, I assume most of you have seen this. I'm just going to do a couple of these real quick just to make sure everybody understands this definition. The author calls this inductive definition. It's really, that's not really right. The word is recursive. Induction is something that you use to, I mean, informally to prove theorems. There's something, you may have learned, if you remember implicit differentiation, maybe from COC-1, that's a special case of a more general theorem called the implicit function theorem. And in this case, these definitions come from a very general abstract theorem called the recursion theorem. So really, when the author says inductively defined, that's not really the right word. It should be recursively, and that's the word that I'm going to use. Okay, so what is A4? Well, okay, so now we have to use this recursive definition, right?  Right? Okay, so what do we want here? Anyone? Six. What's A sub five? Anyone? Eleven. Are you guys getting this? You guys see this? Okay. So the second condition just, again, just says that for N bigger than or equal to four, A sub N is the sum of the three previous ones. That's it.  So now I could easily find A sub six, right?  Okay. Does anybody have any questions about this? Okay, I just assumed that you've probably seen this before, so I wasn't going to spend a lot of time on this. Okay, so, et cetera. Okay, so now what we're going to do is we're going to prove something and we're going to use the principle of strong induction. In fact, what I'm going to do is incorporate another problem that I think I assigned to you. I'm not going to have time to actually do it, but I think I gave you, I think I assigned this. I think I gave you number two in the homework, right? Okay. And I'll tell you what, again, because of time, I was going to write this number two down because we're going to use it in the next problem, but I don't think I'm going to have time to do that. So I'm going to refer to it here in a bit. So let me show you how this goes. So I'm going to, we're going to do another problem now using the strong induction. Okay. And this is going to be a notation as in the previous example. We're going to prove that A sub n is less than two to the n, positive integer n. So here's what we're going to do. I'm going to try to get, I'm sure, yeah. I'll have plenty of time to get through this one. If I have time, I'm going to talk about this Bernoulli inequality problem that you have to do in your homework. I'll help you get started on that. So what we're going to do is we're going to use theorem two. You find the same way as what you were already talking about. Yes. Okay. Yes. So again, second principle of finite induction. It's much quicker just to say strong induction because that's what everyone else calls it. So what is it that we're doing? Well, it's very similar to the problem that I did before with the first principle. So the first thing I'm going to do is just verify the assertion is true for n equals one. And then instead of assuming it's true for n and proving it's true for n plus one, you assume it's true for one through n and then you prove it's true for n plus one. That's the only difference. The strategy in general is the same. It's just that now you have more to get your hands on because your assumption is stronger. So you have more to work with now. Okay. And instead of writing this p of n, I'm not going to write that now. So I'm just going to say, well, if we let p of n be a sub n is less than two to the n, p of one then would just be a sub one is less than two to the first. So that's our base case for the induction in this case. You guys with me on this? You with me? Okay. So again, we're going to... I'm going to label it the same way that I labeled the other one. So one. Okay. And I'm going to be very clear as to what it is that we have to show. We must prove or we must show. I use this word show very loosely here with the base case. A sub one is less than two to the first. All right. That's our base case for the induction here. Well, okay. What's the definition? A sub one is one. So A sub one being less than two then is obvious. Okay. So that's the first part. That's it. That's all you have to say. And as for the second part, okay. Now we're going to let n be a natural number. There are a couple of ways, of course, to do this. Again, I want to be very clear as I said before. I'm not saying that this is the only way to do it. I have to do something, though. So I have to choose one of the ways. This is one way to do it. If you're not sure about your way, if you want to ask as I'm going through this, that's fine, too. And assume, let's assume that the assertion is true. In other words, A sub n is less than two to the n, or A sub i is less than two to the i. For one, two, all the way down to n. Whatever n happens to be. Well, I'm saying it because I'm writing left to right. So I don't mean it from a mathematical standpoint. So what is it that we have to show? So we must prove that it's true for n plus one. And let me just use the same notation I used before. I'll just use the star again. So the assertion for n plus one, then, is the assertion that A sub n plus one is less than two to the n plus one. Is that okay? Okay, so there are a couple of ways you can go with this. I'm going to do it this way because I think it makes my point a little bit better. So what I'm going to do is what we're going to do is we're going to consider a few cases. Case one, so we don't know what n is, right? Maybe n's one, maybe n's two. Who knows? We don't know what it is. So what I'm going to do is I'm going to take care of the easy cases first, and then we're going to assume that n is big enough so that A sub n is defined in terms of the three previous guys. And then we'll use that strong induction hypothesis to take care of that. But remember that A sub n is A sub n minus one plus A sub n minus two plus A sub n minus three only if n is bigger than or equal to four, right? If we're on A2, we don't have three previous guys yet. So we can't apply that yet. We can only apply it when we know that the subscript is bigger than or equal to four because we need to have at least three previous guys in order to use, right? To use in order to get A sub n. So the first case is n equals one. Well, then we're just done by part one because we just verified that, right? We already did that. So we don't have to do that again. We already did it from the top. So let's see. So we have to show, sorry, maybe I was a long clear. We already know that A sub one is less than two. We already talked about that. So we still have to show that A sub two is less than four now, right? That's our assertion when n is equal to one. You see that? When n is one, the assertion is we have to prove that A sub two is less than four. You guys with me on this? So in this case, we have to verify that A sub two is less than four. But this is clear because by definition, A sub two is two, okay? So you should have that in your notes, right? The first three, right? A sub one was one. A sub two is two. A sub three is three. So this is clear just from the definition, okay? These are sort of the simple cases. Case two, n is equal to two. And then what's the assertion we need? We need four. A sub three to be less than what? Eight, right? So we must check that. A sub three is less than eight. Again, that's clear. This is clear since A sub three is equal to three. Okay, so there's another way to think of this. You could sort of think of this as sort of the base case of the strong induction argument, okay? In a sense, okay?  That's only true when n is bigger than or equal to four. So we have to take care of these simple cases first before we know that, okay? Now, the next case is when n is bigger than two, right? Then n is bigger than or equal to three, so n plus one is bigger than or equal to four. And now we can use the recursive definition in this case because we know we're big enough so that we can use it. Okay, so everybody have this down? Okay, so the third case is not going to be that n is equal to three. The third case is just that n is bigger than two. And then that'll take care of everything. So again, I'm going to write down just to be clear. Just to remind you what it is that we have to show, we must prove that A sub n plus one is less than two to the n plus one. Okay, now, by definition, okay, so we have to prove something about A sub n plus one, right? Well, we know what A sub n plus one is in terms of some of the previous terms, right? Again, I don't have this on the board. On the screen, I don't think it's going to help to scroll back. A sub n, this is in your notes, sorry.  So again, like I said, if n is bigger than two, and it's a positive integer, if it's bigger than two, then it has to be bigger than or equal to three, right? So n plus one is bigger than or equal to four. So then we can use this recursive definition then, because we know the subscript is bigger than or equal to four. So it's the sum of the three previous terms. What are the three previous ones? A sub n, A sub n minus one, and A sub n minus two. You guys with me here? Now we can use our assumption, and now I'm probably going to scroll back one just to make sure that everybody remembers this. Okay, let me wait until you're done writing this. So remember what, let's see, oops, sorry. Okay, so one more. Okay, there we go. What are we assuming? We're assuming that this assertion, that A sub n is less than two, and we're assuming that that's true for one, two, three, four, five, all the way up through n. Okay? So we're assuming that what it is that we want to prove is actually true for n, n minus one, and minus two. Everything below n plus one. Okay? So since n is less than n plus one, n minus one is less than n plus one, n minus two is less than n plus one.