Welcome back everybody. Now we're going to wrap up tonight with a talk about spruce disorders. Spruces are one of the most commonly planted trees in North Dakota. It is a great tree. It is so widely adapted. However, it does suffer from many insect pests, diseases, and other types of disorders. So tonight, we've got a leading, if not, I would say the leading tree doctor. If you had a sick tree, this is the person you would want to consult. Joe is the last neck extension for us. I'm quite serious and you will get his email. So just give some more work there, Joe. No, Joe is really, he is the expert when people have a question about how to care for a tree under stress. And we're very fortunate to have Joe here tonight. So let's welcome Joe to the forum. Okay. Well, thanks, Tom. Thank you for that wonderful introduction. It's very humbling. Thank you. All right. We up? Okay. Well, thanks for having me here tonight. I hope I can teach you something about spruce trees and all the disorders they suffer or can suffer here in North Dakota. I do want to point out this presentation is based in part on a recent publication that came out last year. Diagnosing spruce disorders in North Dakota. It was put together by myself and Aaron Bergdahl of the North Dakota Forest Service.  Okay. There it is. Okay. Very good. I'm going to add a few things to this presentation that aren't in here and there are a few very rare or very uncommon pest disorders that I'm not going to cover tonight. There's more information in this document if you're interested. And another document that I'm going to refer to. So that being said, I have diagnosing spruce disorders, mostly spruce. Some of the examples I'm going to give include pine trees because there are some pests that hit both pine and spruce. So we'll cover that here tonight. Just very briefly, there are a couple that are time sensitive. You will see at certain times of the year, we're going to cover those first. And then there are other disorders that you will see just about any time of the year. So very quickly, we'll get into winter injury. Winter injury is number one. Winter injury is kind of a vague term. There are a lot of things that can happen to trees over the winter. It gets awfully cold here and then it gets awfully warm and cold again in the spring. And you combine those factors and it's really tough on trees. So winter injury, it can show up like this. On this spruce tree, this is a photo from Mike Kangus, North Dakota Forest Service. It can show up like this. This is another photo from Aaron Bergdahl, North Dakota Forest Service. So the symptoms you're going to see are pretty variable. The thing is, they show up now. They show up near the end of winter. Actually this winter hasn't been too bad, so we might not see much of it this year. And sometimes there's a pattern. Obviously these aren't spruce. These are pines in the foreground. And everything above the snowpack was killed. All the needles above the snowpack were killed. All the needles below were protected by the snow were not killed. So this is another type of winter injury. Obviously this example is pines. Same on this pine. Very low to the ground where there was snow. Those needles were fine. Above the snow line, I don't want to say toast. I don't want to get too technical, but they're hurting. In this example, kind of more damage on the right side of those trees than on the left. That was more of a wind issue. Or this one could have been road salt actually. Hard to say based on this location. Big pine trees. This is over in Minnesota. And sometimes there isn't a pattern. Sometimes it's pretty random. In this pine, you might see some needles are dead. Some are alive. Some are dead on the bottoms of the branches. Some on the top. So it's really variable. And about the only thing you can do for winter injury after the fact is sit and wait. Wait it out. In this example, you can see where the red arrows are. That's where new growth is coming out. The buds were not killed. The needles were, but the buds were not. So in that case, where those red arrows are, new growth is coming. So you just wait it out. But you'll also notice where the black arrows are, those buds are not growing.  So it's really variable. So that's about all you can do after the fact. Before the fact, actually in the fall, watering from about mid-September until freeze up is what you can do to, I don't want to say prevent, but at least minimize winter injury. Having trees go into winter well hydrated is about the best you can do to prevent winter injury. Obviously, if you have little shrubs, you could use some kind of burlap or other type of material to protect them. Keep them out of the wind, keep the winter sun off them. But for a big tree, you can't do that. The other time sensitive pest I want to mention is Yellow-Headed Spruce Softly, YHSS. What you're going to find here, you need to look at this in early to mid-June, and you need to start scouting well in early June. This pest shows up mostly in Central and Western North Dakota. I've really seen it that far east. I've seen it from Devil's Lake down to Richardson. So everything in between. And with this, you might see twisted needles or half-eaten needles. Or in this case, you could see the needles have started to turn pink and have started to actually kind of shrivel up. So that's one of the symptoms of Yellow-Headed Spruce Softly. On the right-hand side, and kind of towards the background, but on the right-hand side of this photo, you see there's one branch that most of the needles are gone. And that's what the insect does. It can either chew just a little bit off the base and kill the needle, or it can eat the whole needle. And this is a Caterpillar Yellow-Headed Spruce Softly. And here's the thing about this insect. Yeah, that head doesn't look yellow. It looks a little more reddish-orange to me. Yeah, sometimes they're like that. However, the interesting thing with this is it's really not a tough Caterpillar. It's pretty sensitive to the chemical, to insecticide control. It's very easily controlled by insecticide if the timing is right. You have to have the insect there. You have to have a window for about four weeks, look for it, find it, scout. You have to be out there and find it. And then, yeah, most insecticides will take care of it. But if you spray too early or you spray too late, that's just a waste of money. So that is Yellow-Headed Spruce Softly. Keep an eye out for that in starting in early June. Okay, oh, and there we go, control. There we go. It's everything I just said right there in two lines. Timing is critical. As we move on, we're going to talk about spruce disorders. You're going to see it any time of the year. And I wanted to add a little caveat. Any time of the year for diagnosis. Okay, you're going to see these a lot, different times of the year, but the treatment is often time-sensitive. There is an insect pest that you can see this every month of the year. It's always there. Treatment for it is really time-sensitive. We'll talk about that. But first, we'll talk about the needle cast diseases. Needle cast diseases are going to show symptoms like this. See, the bottom branches are bare. The interiors of the tree is bare. If you look at an individual branch, you might see something like this, where the most recent year of growth, they're on the right. The last year of growth, the needles look fine and then you go in one year and the needles are still there. You go in to the third year needles, they're mostly gone. Fourth year needles are totally gone. So you need to look at the different age class of needles. The first year should always be the second year, second age class there. They might already be affected. If they're infected, excuse me, that's going to be really tough on the tree. Third year and older, that's going to be tough as well. Spruce trees should hold their needles for three to seven years. Sometimes, and I would say closer to that, seven. Usually I'd say five to seven. But they should hold several years worth of needles. And if they're losing them, that could be a problem. So the lower needles are more affected, as I've said. But here's the question. Very simply, how can you tell the difference between a needle cast disease and just simply the effects of shape? As trees get older and grow bigger, those older needles towards the center of the tree get shaded out. That's why the tree lets them go. How can you tell the difference? You can tell the difference by these fungal fruiting bodies. These are on the needles. These are photos really close up of the needles. And those little dark dots are fungal fruiting bodies. On the right is a fungal disease called rhizosphera. On the left is one called stigmina. And it's really hard to tell the difference. We'll talk about that in a moment. Normal needles, all those stomates. Stomates are the little pores in the needles. They're just nice little lines. They should be white. They should be white. You can see in those bottom photos, some of them are white, but mostly they're black. Those are the fungal fruiting bodies. That's how you tell the difference between needles that just got shaded out because they're not going to have those black fruiting bodies. The way you tell the difference between these two diseases, it's very simple. Stigmina, those fungal fruiting bodies, they're kind of fuzzy edges on them. Obviously, you can see it in this photograph. With a 10-power 10x hand lens, you should be able to see it. It might be a little tough. You might need to get it under a microscope, but if you see those fuzzy edges to those fruiting bodies, that's stigmina. Rhizosphera, they're smooth edges. Over the last few years, it seems like stigmina has kind of kicked out rhizosphera. Rhizosphera used to be the most common of the needle cast diseases, but it seems like stigmina has kind of taken over and replaced it. In terms of control, rhizosphera, it wasn't easy, but it was pretty straightforward. Chemical control two times early in the season. First is when the needles are about half elongated, and the second would be three to four weeks later. As a rough rule of thumb, it's Memorial Day and Fourth of July. Rough rule of thumb. As we move more towards this stigmina, it seems like that four-week, oh, and sorry, let me back up with rhizosphera, that two-time treatment, Memorial Day, Fourth of July, you would need to do that in two consecutive years, and that would work to prevent future infections, or at least minimize. It's really good at that. Stigmina, it's kind of an ongoing battle. Yes, there are those two applications, but it has to be continued throughout the growing season for three or four or five applications, depending on the weather, and several years in a row, more than just those two years. We really need more research on it, but that's our recommendations right now based on our experience. It's a tough disease to treat. We find this mostly where it's wet, I guess in the Devil's Lake area, all over the place. As you get further west, it lessens, it tends to decline. West of 83 or 85, that goes from Bismarck to Minot. 83, I always forget. West of 83, I've seen it a few times. It's not nearly as common as it is east of there. If you're looking for more information about this, our extension publication, F-1680, has more information about the needle cast diseases. Another one you can see just about any time of year is Volsa canker. We used to call it Cytospora canker. We're going to be in this transition mode for about the next 10 years because we've sold Cytospora canker for so long. It's called Volsa now. This one is interesting because it's just individual branches.  That's one of the key diagnostics, but not the key diagnostic because a branch can die because of a lot of different things. We just have had too much snow on it, for example, and it broke. Just physical cracking could have been, I don't know, squirrels eat branches sometimes. That's much more so. Here's an example. On the left, that tree, I took that photo in 2005, the branch on the left of the left photo with the green arrow, the color was a little bit off and it caught my eye, so I took that photo and you can see a year later in May 2006, the entire branch is dead. Now, also on that tree on the left, in the red arrow, pointing to a branch that looks really healthy and is really healthy. A year later, totally gone. Well, jeez, what's happened there? The key diagnostic for Volsa canker is these fungal fruiting bodies that you find just under the outer bark, just in the inner bark. That's the key diagnostic. And it's really hard to see sometimes. Basically, treatment is very simple.  And keep the trees properly watered. Volsa is a really interesting fungus in that it's really associated with moisture. If the tree gets drought stressed, if a spruce gets drought stressed, it's very sensitive to this disease. The flip side of it is don't overwater tree, overwater spruces, because they're pretty sensitive to flooding. See, it's hard to talk to an online audience because I'm so used to say, any questions out there? I know we're saving questions till the end. They're coming, Joe. All right, lots of information. I had spruce spider mites. You see needles like this, where the needles get a mottled appearance, mottled appearance, where they, you're usually green, but there are these patches of yellow or tan. You'll notice, again, this is going back to another issue. Stomates, the stomates here are all white. There's no needle cast on this. And the spruce spider mites are tiny. Oh, my gosh, they're small. This is, I'm so glad Erin got this photo of that insect. So it's a cool season mite, about, about 80 degrees, it goes dormant. So it's active early in the season and it's active late in the season. Incredibly small, hard, hard to see. Culture control works pretty well. Just spray a jet of water on there and knock them off of the chemical control. They are mites. They are not insects. They're more closely related to spiders.  However, they'll also kill predatory mites. Some mites will eat plants and some mites will eat other mites and insects. So, gotta weigh that. Timing is critical. Again, it's an early, it's either early season or late season. And get out there and try to find them. An easy way to do this is take a white piece of paper and put it under the branch, shake the branch, and see if there are little black specks moving around on the piece of paper. I've also heard of the smear test where you get those black specks and you run your finger across them and see if they smear on the paper. Crude but effective. Okay, spruce spider mites. Pine needle scale, this is one of those you're going to see at any time of the year. This is an insect pest that hits pine trees, as you can see in the photo on the top, but also spruce trees. I've seen them right next to each other in the same shelter belt, a pine tree and a spruce tree and they both have pine needle scale. It's a tiny insect that you see those white coverings any time of the year. The eggs are under there. The mama scale has created that safe zone for the eggs to hatch. For the eggs to hatch, you can create the crawlers and the crawler stage is from May to early June. So again, timing is critical. Yes, horticultural oils may work, insecticides are many that will control scale insects, but again, timing is critical. You have to be out there when the crawlers are out. That's when the chemical has to be applied or the treatment has to be applied. So scouting, scouting, scouting is important. Okay, a few more I have listed as other. These are more environmental things that may or may not actually be in our diagnostic guide. This is a pine tree. In this example, it was rodent damage. It was voles that killed this tree, but I've also seen where the whole tree dies because of a line trimmer has pretty much girdled the tree. So what do you do in situations like this? You cut the tree down. There is nothing you can do, unfortunately. But if the whole tree goes all at once, then check at the base of the tree. Check right on the stem where it hits the ground and see if it's been girdled. Another thing that might happen is the site. For example, in 2012 in Bismarck, this tree is kind of on its way out, unfortunately. And they're sitting right in a pool of water. 2012 was a big Bismarck flood, just like these trees. Some of them are already dead. One of them is not. People say, well, why didn't that tree die? Who knows? Genetics, maybe that one's just a little more resistant. It's a little bit bigger. Maybe it's age. Maybe it's a site thing, but I would be willing to bet that maybe that tree lasted a little while longer, but I'd be willing to bet that it's dead. For the areas in Minot that were flooded in 2011, you'll be hard-pressed to find any conifers in that area right now that survived the flood. There are few, but there are few and far between. Okay, here's another example from Kindred, 2006. My old boss said, hey, he left a branch, a spruce branch on my desk. He said, hey, Joe, what's wrong with this tree? It's killed this one tree and it's going down the line. It's a dead branch, I didn't know. So I went out to the site, and this is what I saw in the first tree. And there it is. If you go to the left, that tree is in decline somewhat. And if you keep going down the line, well, further away, the trees are healthy. Okay, so yeah, he was right. It's starting at one end, going down the line towards the other. And I go back to that first photo, and I knew exactly what it was right away. Look at the house behind it. The house is about four to six feet higher than the driveway. And that house was pretty new. So what happened was they built a new house. They graded the ground and put in a lot of fill soil. Okay, spruce trees cannot handle fill soil. They can't handle flooding either. It's a low oxygen environment. So that's what was killing the tree. Yes, insects came in and probably some disease and finished the tree off. But really, it was that fill soil that caused the initial decline, the initial stress that allowed these other pests to come in and kill the tree. Okay, that was other. And now we're into more other. These are some smaller trees I saw in Fargo. I'd say in 09. And where I have those arrows on that individual tree, there's something different between the top of the tree and the bottom of the tree there. The bottom, it looks more full, a little more green. Above those arrows, it's a little more yellow and a little more thin. It's a subtle difference. It's a subtle difference. You have to have the artist's eye to see this. And, you know, in the background, I've got sky in the background of the photo and lawn in the background. And it's hard to see, but there is a subtle difference. Below those arrows, it's fuller. It's darker. Above is more yellow and a little more open. So I looked at the stem right in that area. And that's what I found. Okay, the strap was still on the tree. Many years later, it was choking the tree. It was girdling the tree. And there was a big swelling above that, where all the food made from photosynthesis was piling up. And I went back a year later to go check on those trees. They were gone. So, what can you do? But along the same lines, here's a big, big spruce tree where, same thing, same idea. That tree's enormous. There's something happened on the stem at about that height. In this case, it was sap suckers. Blew me away. Couldn't believe it. A sap sucker is a bird. You cannot kill it legally. Do not. They are a type of woodpecker. They're not going after insects. They're going after sap. There's not a whole lot you can do about sap suckers. And unfortunately, we always see this damage after the fact. I'm working on my own apple tree at home right now. Those buggers took out my apple tree. So, trying to figure out how to fix that. Okay. And that's going from more other to more extra. A couple of other things before we finish up here. Herbicide damage, herbicide damage on spruce trees is sometimes hard to diagnose. Sometimes it's pretty clear. We often see damage on conifers going up the tree and then a kind of a spiral. That's the way water goes up conifer stems is actually spirals up the stem. Other trees are different how water moves up the stem. Well, if herbicide is in that water, it will spiral up and kill branches along the way. And sometimes it takes an artist's eye to see that spiral. Sometimes it's hard to see. Sometimes it's not. In this case, we don't know what the causal agent was. This was over in Moorhead. There was a railroad siding there and some kind of herbicide was sprayed to keep down the vegetation. And what happened is all the spruce took some of that in and it turned those outer needles yellow. Here on campus, NDSU campus and that fence in the foreground, somebody was spraying to keep down the vegetation and I never quite figured out what the chemical, what the active ingredient was that was there. This was in Headinger a few years ago and we tried to figure out what was sprayed. Someone said it was roundup or glyphosate. Someone said it was 2,4-D. But we were never quite clear. In this case, the color was interesting. It was more of this purplish pink. But what we noticed besides the purplish pink is you'll notice that it's always the branch tips. It's the active growing points or most recently active growing points where you're going to find that damage. In some cases, it starts to twist the needles. There's also a pest called a phytoplasma that may twist the needles in a similar way. We're still trying to figure that one out as well. In this case, with all the other herbicide damage, we're pretty sure that's what caused the damage here. That being said, I wanted to go through that fairly quickly. There's a lot of information there. I'll be happy to do my best to answer some of your questions. Thanks. Thank you, Joe. We've got some questions here for you. One is, deer are browsing on a mucle pine. The needles are gone, but the buds are still there. Will the branches survive? The branches should survive. If the buds are there, that's where new growth is coming from. Hopefully, there's enough new growth to make up for the amount of needles that were lost. Again, you're just going to have to wait it out and find out. I think, well, they should be okay, but we'll see what happens. How about that spruce tree? Will the top got broken off? Will the spruce tree still grow? Yes, it will. The spruce will still grow. However, with the top broken off, it's hard to say. If it's a small tree or if it's just the very tip, the very leader, like one or two years worth of growth, a side branch can be tipped up either on its own or you can help it along. If it's large, two, three, eight inches diameter, this side branch eventually will make its way up to become the leader or actually several side branches, which is going to create a point where structurally it's going to be weakened. If it's something you can direct a new leader, great, do that. If not, well, be careful in the future. Joe, how about the difference between the rhizosphere and the stigmina fruiting bodies? Is there a way to tell the difference between the two? Sure. Yeah, stigmina, it's subtle, but you can see it under a microscope or with a 10-power hand lens. With the stigmina, the edges of that fruiting body are going to be fuzzy. They're going to be blurry.  With rhizosphere, it's pretty much round without the gloves. They're both those fruiting bodies are going to be linear on the needle. In those stomates, yes. They're laid out in a line. You have to look at those black dots carefully, like Joe said. Okay, let's say they have stigmina or rhizosphere. Do you recommend a fungicide for that? Yes, fungicides with the active ingredient, chlorothalonil. Yeah, I guess I should have added that. Great question. Chlorothalonil. There are a variety of products that have chlorothalonil as the active ingredient. Look those up. How do those needle cast diseases spread? The needle cast diseases spread. The spores get released, I think throughout the year, but mostly it's early in the growing season and they get spread. If the new growth is susceptible or receptive, that's when they can get infected. It's that new growth that's still soft and tender that is most susceptible. Later in the growing season, they're hardened up, they're less susceptible, less likely to get infected, although it can still happen. Okay, if an ice storm breaks off the branches on one side of a tree, will that side of the tree ever grow back? If an ice storm breaks off the branches on one side of the tree. Well, probably not. But that being said, sometimes trees will put out new branches off the stem. I've seen where if trees are heavily, heavily pruned or heavily, heavily stressed, they will send out new branches off the stem. It looks a little weird, but it can happen and it can grow. Yeah, for a long, long time. For a long, long time. It may take a long time to fill in. Okay. What's the latest verdict about using Roundup at the base of a tree trunk to keep the grass awake? The latest verdict. You know, there's some research that's... Oh, there's no easy answer yet. Sometimes Roundup does get into the trees. Sometimes it will get into a root or into the stem and it can cause some damage. Sometimes it doesn't. Be careful. Use it properly. Use it according to the label. Try to keep it off the tree stem if you can. If there's a wound on that tree stem or on a root, it'll get in from that wound, that fresh wound. So, try to keep it off. So, how about... Does a copper fungus spray work to get rid of black fungus growth on a small, new, blue stem? Blue spruce totem. This spruce totem, like a totem pole. I don't know. If you see black fungus growth on a young blue spruce, what do you think that is? A big, sweaty mold, maybe? It could be. Fungicides are almost always preventative. They're not curative. So, in cases like that, after the fact, it's probably not going to do much. It might prevent new infections and that would be a way to go. I'm also going to say at this point that it's hard to answer questions, because usually when you do a diagnosis, if not just like, give Joe 20 some words and here's your answer. Photos help. So, yeah. We are in the age of a digital camera and it can make such a huge difference. So, take a series of photos, close-ups, distant photos and send them to your county extension agent. They're there to help you for any type of tree disorder and then Joe is also there to support the county agent and to support you to give you the answers you need and in our digital age you can get an answer rapidly and you get research-based information from NDSU Extension. We won't try to sell you a fungicide that you don't need, for example. So, I don't know if you can do this one. Can you use well water at a pH 8 sodium level 800 ppm to help establish spruce or pine? That's a pretty specific question, but how about maybe you can talk about in general about the salt tolerance of spruce or pine compared to most other plants. You know, salt tolerance really, there's not a lot of plants, there's not a lot of trees that are salt tolerant and using water like that I would say you could use it in an emergency, but if you can get better water than that, that's a pretty high salt level. I'm not so worried about the pH. Yeah, our pH of our water is a little high. The pH of our soil is a little high. The spruces that we have here which are Colorado blue spruce, Black Hills spruce, Meyers spruce a little bit and in urban areas you see Norway spruce, they do tolerate our higher pH soils fairly well. Okay, how about do you know of any public tours of trees at Ebseraka? Just pay attention to the field day. Yeah, pay attention to the field day, August. Yeah, really August. Yes, every year is a field day and actually the people you had here tonight are the people who will leave those tours. When you prune for Volsa canker, do you prune the whole branch if only part of the branch is affected? Or just prune the part that's affected? Prune the part that's affected. If it's a sub branch of the main branch yeah, you can take out just the dead area that if that doesn't work, then you can go all the way back to the main stem. How about a chemical name to control pine needle scale? Pine needle scale, there's horticultural oils. But there are a lot of insecticides that will control scale if the timing is right. You know, a metacloprid as systemic. They care. It is early spring, yeah. Metacloprid as a systemic does control scale, but under the right circumstances carburel carburel which is also systemic can be sprayed thickened and malefiant malefiant will work. If you get it when the crawlers when the crawlers are out at the right timing, yes. Okay, a general cultural question. They have several 20 foot tall Colorado spruce trees planted in late September. Should they fertilize them in spring? I'm not a big fan of fertilizing trees in North Dakota. Generally, there's plenty of nutrients in our soil. If you are going to fertilize spring is the time to do it. But really the most limiting factor here in North Dakota is water. You know, that's the biggest issue. Far and above the nutrient issue. And 20 foot tall Colorado spruce planted in late September. Boy, I hope that had a big root system as part of it. Re-establishing that root system is really critical and water is going to be the critical component to get those roots re-established. Okay, here's something that's a really useful question. The idea that cutting off the leader is going to make your spruce tree fuller. Is that true? Do you recommend doing that? I don't recommend doing it. It can make it bushier. You know, when they share Christmas trees, spruce trees, yeah, they will clip the leader and clip the leaders of the branches. With that being said, boy, I wish I was able to do, have some diagrams here. Spruce trees are interesting in that, can you see my finger? Okay, buds on spruce trees are all throughout that leading branch. So if you clip just the top off, there should be other buds below that will take over. That will kind of compete to be the leader. So it'll make it bushier, but what you'll lose, and that's okay if you're growing a 7 or 8 foot or 10 foot Christmas tree, but what you'll lose in the long run is you'll lose a central leader. You'll have multiple leaders in years down the road. That's a bad thing. So we generally do not recommend sharing your landscape trees for spruces. How about this long-care company said that they can spray something on their spruce tree to help protect the tree winter. Okay. What do you think about that? There are products called anti-transference, those pores in the needles, those stomate pores. That's where a number of things happen, but that's where trees lose water and an anti-transference, it's called transpiration when they lose water. So an anti-transparent is basically a waxy coating that plugs up those pores. They've been kind of hit or miss here in North Dakota. They work for about three months. You have to apply them when temperatures are above freezing. So you apply it in October. So three months later is November, December, January. January generally is not above freezing in North Dakota. If you can time it right, find that mid-winter thaw. You might be able to do it and it might help, but it's going to be tough. Okay, Joe. That is going to be it. So thank you, Joe, for that outstanding presentation. And all of it, sorry, you got one person. How do you get rid of spiders on a Bruce Bruce? Spiders are spider mites. I'm okay with spiders. If it's spiders, they're beneficial. They're eating the insects. Don't worry about that spider mites. They're those mites. Jet spray of water is the common way to control it. Just use a jet spray of water in this case. You can go. Okay. Thank you, Joe.


If I had used good judgment, I wouldn't have taken this job in the first place. But now that I've come this far, I'm willing to go on just to see what else will happen. Have gone. Will travel. Starring Mr. John Daener as Paladin. San Francisco, 1875. The Carlton Hotel. Headquarters of the man called Paladin. No, hey boy, you're interrupting a very important game. You play cards all by yourself? I play solitaire all by myself. Oh, solitaire. He's game of luck. Ah, he's game of luck that becomes game of skill in the proper hands, hey boy. Wait a minute, wait a minute. I bet you can't beat it. Who may I ask are you? Gentlemen, gentlemen, please, please. One at a time.  Now, hey boy. These gentlemen come to desk, Mr. Paladin. Ask if I know who you are.  I see. And you, sir? Well, my name is Howard, Mr. Paladin. Jamie Howard.  Well, this particular game has been temporarily suspended, Mr. Howard. At least until I find out just what your business might be. My business? Well, yeah. Oh, yes. Well, you mean why do I want to see you? That's what I mean. Well, sir, it is about my gold mine. I lost it. Yeah, I, I heard about you and I figured maybe you were the man to help me get it back. You lost a gold mine? Well, not exactly. I know where it is, all right. I say I know where it is, but when I began to hit pay dirt, a fella swindled me out of it and had me run off my claim. I see. And just where is this gold mine? Up back at Columbia, near a place called Yankee Hill. Columbia? Those mines were worked out years ago. Oh, give me five to one, they weren't. Not all of them. No, sir. Here, take a look at this. That's gold, all right. I bet your life it is. That's why they stole her. I snuck back and took these out of the mine after. I mean, it figured maybe you wouldn't believe me. Oh, I can pay you all right. I can pay you plenty once I get my mine back. I see. Mr. Howard, I think you have just made a deal. Well, good. I'm thankful that you're willing to help. Well, hey, boy, what do you think? Oh, Mr. Paladin, why don't you play Black Seven on Red Eight like Mr. Howard said? Constipation can be a problem for anyone, even doctors. And when constipation occurs, it's interesting to see just what doctors consider important about a laxative they might use or recommend. Well, a majority of the doctors we heard from had this to say. A laxative should be effective, gentle, close to natural acting. A medicine that can be used with complete confidence. Now, x-lax has been popular with many doctors and millions of people over the years because chocolate and x-lax is effective. Overnight, it helps you toward your normal regularity. X-lax is so gentle, so close to natural acting, there's no upset. That's why many doctors and millions of people use x-lax with complete confidence. X-lax, the laxative that helps you toward your normal regularity. Gently, overnight. We got an early start the next morning, early, but not altogether bright. At least not for grizzled old Mr. Howard. He looked like he'd spent the night taking in the sights of San Francisco. I'd seen his type before. Prospectors in from the hills looking strangely out of place among the gas lights and horse cars of a modern city. We had a long ride ahead of us and I saw to it we covered as much ground as possible. Howard didn't have much to say until about noon when we pulled up alongside a little stream to let our horses drink. Oh, my doggy. I'm glad to rest a spell, Paladin. Well, the horses are too, Mr. Howard. If we're going to reach Columbia, we can't run them into the ground. You know, hey, you hear that? Oh, hear what? That bird. I'll give you treaty one, that's a Papa Robin bringing some food home for the youngin'. You will, huh? Yeah, you want a bet? Nope. You better save your gold, Mr. Howard, you might need it. Yeah. Say, I meant to talk to you about that. About what, Mr. Howard? About that gold. See, I don't have it anymore. You don't have it? Well, Mr. Howard, there must have been close to $700 worth of gold in that pouch you showed me last night. How have you been going through that in one evening? Well, sure. Now, I'll tell you. I'll tell you. I met this fella and we got to talking about poker. He decided to play a little and the first thing I knew, he had it and I didn't. Mr. Howard, I ought to turn around right now. No, no, no. Don't you worry now. Don't worry. There's plenty more where that come from. Soon as you get my mind back for me, we'll be rich, both of us. But just how much ore do you think there is in your mind? Well, it's hard to say. But it should take out $600,000 worth, easy.  Well, sure. It doesn't feel good to do a little gambling down there. It makes life interesting. It's interesting. That's how I found all their mother ones. Taking a gamble where others wouldn't. That's how you found all the other what? Why, all the other mind. Well, son, I've been through four or five fortunes for this one. But, well, I'm getting older now and that's why I'm hanging on to this one. Yeah. Well, I'll be better moved on.  Come on. Mr. Howard, how long has it been since the country around Columbia saw its last big boom? Oh, about 10, 12 years, I guess. Of course, there's still a few of us around, figured that there was still a strike to be made, and I proved it. Well, then tell me, isn't there still a sheriff in Columbia? Oh, sure there is. Farrell's his name.  When I told him? Yeah. Sure, I didn't have to tell him. He already knew. He did? Oh, sure. He was the one that runned me off. He was the... Who? Uh, Mr. Howard, did you say the sheriff ran you off your claim? Well, now, wait a minute now, Paladin. Say, wait just a minute. I guess I should have told you. Sheriff Farrell, he is in cahoots with the other one. What other one? The other Farrell, the one that runs the minor's friend Saloon up there. A couple of dirty, black-hearted scoundrels. And they're related? Well, sure. Sheriff's a brother to the other one. Well, all right, Mr. Howard. I don't know why, but I'm willing to go through with this just to see what happens. Oh, thanks, Paladin. It's just one other thing. Oh, something more you haven't told me. Well, he suggested I've been thinking about this Mr. Howard business. Would you mind just calling me Jamie? Everybody else does. Okay, Jamie, it's a deal. Yeah, I don't know, Paladin. Might be we shouldn't have rode right into town like this. Right down the main street and all. You just let me worry about that, Jamie. Yeah, but suppose somebody sees this. Jamie, I want them to see us. All right, you two. What'd I tell you? What'd I tell you? That's the sheriff. Oh. Just get them hands up and turn around. Slow when he's dropped your gun belt right where you're beating. No, just a minute. Is this the way you welcome a visitor to town? Visitor? I don't consider no partner Jamie's a visitor. You better do as I say. All right. That's good.  Sheriff, why are you doing this? From what I hear, you've already run him off his claim. Isn't that enough? That was then. Now I'm arresting a pair of horse thieves. Horse thieves? Well, now that there was another little thing, I guess I should have told you, Paladin. This here horse, it belongs to the sheriff. But Jamie, why would you steal the sheriff's horse? Why? Well, everybody knows he's the best one in town. If I'd have taken any other horse, they'd have caught me before I got five miles away. That's enough talk. You just start walking. I'm taking you to jail. Both of you. Whenever you go this summer, go first by long distance. Excuse me? Can you tell me where the conleys live? Yes, in the White House, at the end of the block, on the right. But I'm not sure they're home. Oh, we called them long distance. They're expecting us. Thank you. A long distance call to friends you plan to visit this summer makes good sense. It will save you driving miles and miles, then finding no one at home. Call ahead to make sure of reservations, too. Wherever you go, go first by long distance. Rates are low. Why not call now? Oh, Shaw. I'm sorry about all this, Mr. Paladin. You should be, Jamie. Yeah, but I'm not worried, Nun. No, sorry. You're not? No. I say I figured you for a man that can get us out of here pronto. Well, thanks for the vote of confidence. Well, anything I can do to help, just let me know. No, I think you've done just about enough, Jamie. Well, it isn't like I really stole the sheriff's horse, you know. I brought her back, didn't I? He's got her tied up in the stable out in back right now, ain't he? Yeah, you may have a point there, Jamie. Sheriff? Hey, Sheriff! Stop that noise. Have you know I run a peaceable jake? Sheriff, I think maybe we can make a deal. The only deal you'll be making is with a rope. After a fair trial, of course. Horse Theories are hanging off in these boxes. Listen, listen to me for a minute. In a way, you're responsible for all this. Did you or did you not run Jamie off his claim? Oh, sure I did. It was my legal duty. That's what I'm paid to do as sheriff. But that didn't give him no right to take up horse stealing. No, wait a minute. You have your horse back, don't you? Sure, after caught the thief. But you didn't catch him. Jamie brought the horse back of his own accord. Well... That meant that you were without the use of your horse for five, six days, right? Five days. Now, ten dollars a day seems like a fair rental. Supposing you had rented your horse to Jamie. But I didn't. He stole it from right in front of the jail. I said supposing. Now, five days at ten dollars a day is fifty dollars. Now, if Jamie paid you that and threw in an extra ten, wouldn't you be, well, willing to call it square? Well, now, that might put a different picture on it. It'd cost ten dollars to ask someone to build a gallows anyhow. Jamie. Eh? Pay the sheriff his sixty dollars. Pay... Well, Paladin, I ain't got sixty dollars. Like I told you, I lost my gold in San Francisco. But once I get my mind back... Of course, I guess I could use you prisoners to build a gallows yourself. Hold it, hold it, Sheriff. I've got sixty dollars. It's in my saddlebag. Hey, this is a mighty nice room. Oh, I didn't know this hotel was so nice. All right now, Jamie. Before we go any further, I want you to know something. I've got sixty dollars invested in you. If it weren't for that, I'd turn around and walk out that door right now. Oh, don't you worry about that, Paladin. You'll get your money back. And plenty more once I get my mind back. Listen, Jamie, just what is the true story? The sheriff said it wasn't yours. Oh, it's him pharaohs. It's him pharaohs. They're all alike. The sheriff and the other one that runs the saloon. They're crooked games. What crooked games? Why, damn, card games with a crooked dealing. Oh, I've seen that dealer slipping cards off the bottom of the deck. I know. And then when they found out from the man at the essay office just how rich my strike really was. That's how you lost your mind? In a crooked card game? That is how I had it stolen. They made me sign it over to them.  That's why I come for you to get it back for me. But, Jamie, if you knew the card game was crooked, why did you get in it? Why didn't I get in it? Well, now that is a silly question. Caught her a new other place around here to gamble. That's why. Somehow I had to get away from Jamie Howard. I went downstairs and started to walk up Main Street. And then I noticed across the street the miner's friend saloon, F. Farrell, proprietor. I decided to talk to Mr. Farrell about those card games with his. I haven't seen you around here before. No, I'm afraid you haven't. Well, let's make up for lost time. How about a drink? Later, maybe I'm here on business. Oh, well, maybe I can help you. You can. You can just tell me where I can find Mr. Farrell. Mr. Farrell? Well, the man who owns this place, the man whose name is on the sign out front. And just what is so funny? There isn't any Mr. Farrell here. I'm the owner of this place and my name's Floss Farrell. Oh. Now, you wanted to talk business? Yes, yes, I did. My name is Paladin, Mr. Farrell. So? I'm here to represent Jamie Howard. Hmm. Well, Jamie, I haven't seen him around for a while. He's been living kind of high since he struck it rich, hasn't he? But that's all over now, isn't it? Just what you mean? I mean he's no longer living so high since you took his mind from him in your card game here. Oh, not me, Mr. I don't run any game. I lease the gambling concession to Bill Hogarty. Oh, where would I find him? Back at the poker table. He's a fellow dealin', the big fella. And now, here are Edgar Bergen and Charlie McCarthy. Charlie, where have you been? Well, it's a long story, Durgan. Remember, you said you wanted the car lubricated. I said I wanted a guardian maintenance lubrication. Like all Chevrolets, Pontiacs, Oldsmobiles, Buicks, Cadillacs, and Chevy and GMC trucks, our car deserves the best of service. And that means guardian maintenance at our dealer service department. Yes, well, I drove our car to our dealers without an accident. Oh. Most of the way. Charlie, where's the car now? It's on Main Street between 4th and 5th. Is it closer to 4th or 5th? It's all the way from 4th to 5th. All right, young man, you're going to get it now. Ha! I'm only kidding, Durgan. It's just a scratch fender. Our dealer's GM train mechanics have already got it looking like new again. It's part of their quality appearance service. Shall I still meet you in the woodshed? You drove the car without permission. It's the woodshed. And I'll see you there as soon as I phone the dealer. Well, take your time. If you're not there in 10 minutes, I'll start without you. I took a seat in the poker game. Bill Hargety was dealing and watched him work. He didn't look like the typical card sharp. He was big enough to scare most men by his size alone. The livid scar along his left cheek didn't do anything to gentle his appearance. But he didn't need all that. He had a gambler's hands. And the cards came out of the deck for him like they were trained. Off the top. And off the bottom, too. Well, that's too bad, boys. Seems as though luck just isn't running your way tonight. Deal you in again, Mr. Paladin? Yes, Mr. Hargety, if you don't mind. I have a hunch my luck may change this hand. Well, that's the way to think, all right? Or... Yes, sir. There you are. All right. I think I'll just open for... 50. 50, huh? Well, I'll match that and raise you 100. Oh, I'm out. That's too rich for me. Me, too. How about it, Paladin? You still in? Still in, Mr. Hargety. I think I'll have just as much luck with these as any of you might slip me off the bottom of the deck. What do you say, mister? I'm just wondering if that was how you managed to win Jamie Howard's mind from him. You accusing me of cheating? You've been dealing cards off the bottom ever since I sat down at this table. I'm not wearing a gun, Paladin. In a few words... I'll be happy to leave mine right here on the table, Mr. Hargety. There. Now? Now! You got him, Paladin. Here, here, you got him good. Jamie, where'd you come from? I was going to Mr. Fun, did you? You were right about the game. It was crooked. Let me have that beer, will you? Here. All right, Hargety, you want any more? No. No, I had enough. Where's the deed to Jamie's mind? There you are, Jamie. While Hargety had the deed, he was within his legal rights to have Sheriff Farrell run you off the claim. I'll see that he signs it back over to you. I'll take that paper if you don't mind, Mr. Paladin. You what? Yeah. Might as well give it to her. What's this all about? Well, I say, well, Paladin, I'll tell you now. You'll tell me what? What he's trying to say, Mr. Paladin, I offered him 10,000 cash against his mind that you'd win the fight. That... And you bet against me? Well, Paladin, that was an awful good bet. Oh, look at the size of him laying there. Jamie. Well, now, don't you worry nothing about your money. Jamie. You'll get it. You'll get it just as soon as I find me another mind. Jamie. Oh, I give up. Oh. Yeah. Who is it? That's a pleasant thought. It's good to wake you up in one. Oh, Miss Wong, this coffee is strong and hot enough to wake up old Rip Van Winkle. Just an acquaintance, Miss Wong. Hey, boy, say you get back late last night. Very late. Oh, hope trip. Very fortunate and lady luck right by your side. Miss Wong, the luck that rode by my side was far from being a lady. Pepsi Cola refreshes without filling. Why? Because it's truly light. Charlie, you're forgetting something. Wait, Kay, there's more. Yes, ice cold Pepsi is the delicious refreshment that goes great at a picnic or a party. But, Charlie... And Pepsi goes fast. People like it, so keep plenty handy. There. Oh, you did fine, except for one thing. Well, I mentioned lightness and how Pepsi refreshes and how fast it goes. You left out Pepsi sociability. You know the B-Sociable song. But, Kay, I can't sing. I can. Listen. B-Sociable looks smart Keep up to date with Pepsi Drink like refreshing Pepsi Well, at least I can say this. Pick up an extra carton of Pepsi today. Please do. This is Hugh Douglas inviting you to join us again next week when CBS Radio presents Have Gun Will Travel


This is a day in my life as an events intern working at the Walt Disney Company. I'm studying a Bachelor of Commerce at Curtin University and majoring in event management and digital and social media. This morning I did a quick workout before coming home to get ready and head to work. We then had a site visit to the Riverley Cinema. We'll be having a screening event here for one of our movies later this month so it's really important we get a good feel of the venue beforehand. At about 10 a.m. we headed back into the office. So this day I was working on some write-ups for some previous event screenings and I also finalized an agency briefing for an event we have coming later this year. By this point in the day I was starving so we went out for some lunch. This was my little outfit of the day. I worked on some event budgets in the afternoon and then it was time to go home. I ended up grabbing some pasta for dinner and then went home, watched some Netflix and went to bed.


Today I'm going to teach you how we do syllables. Syllables are words that are broken into chunks. We have one syllable words, two syllable words, and three syllable words. I have today a ball and a cube. This is a one syllable word and I'm going to show you how we can teach syllables. Children love movement. They love to move, they love to learn. So I'm going to do the clapping syllable. Then I can also do the stump, stump, stump syllable. I can also do the chomp, chomp, chomp syllable. But I'm going to start with ball. So the first syllable, let's do the clapping syllable for ball. Ball. I'll ask my students how many syllables. And they'll have to say one, cube, one syllable. So they love it. Sometimes they hear two. Then I'll say, okay, listen. I'm not going to say a word. I'm just going to clap it out. And you tell me how many claps there are. So I just clap out. And then they get it. They say, okay, ball, one syllable. Then we move on to two syllables. I broke it down so that you can see how the word, the sub words are broken up into syllables. These are two, a, po, ye, lo. Now let's use the stump, stump for these two syllables. So I'm going to say, a, po. And my students will say, I'll say, how many stumps did I have? And the students will say, two. Ye, lo, two. So one syllable, two syllable. And then we get to three syllable words. Three syllable words. Try, and go. Try, and go. Sur, lend, da. Sur, lend, da. Let's use our next movement, which is the chomp, chomp movement. So let's do triangle first. Try, and go. How many syllables? Then my students will have to say three. So let's try the next one. Sur, lend, da. And I'll say, how many syllables is that? And they'll say three. And that's a great job. You guys are doing a great job. What they love most about the syllables is their names. So I'll use me as an example. I'll say, I am Mrs. Dennell. So how many syllables are there in my name? Let's count. Dan, no. How many syllables is that? Two.  We can learn about syllables in various ways. We can do the months of the year like January. January has got four syllables. Now this is very tricky for young children. They got to listen very carefully. So this is when I say, listen carefully. I'm going to clap it out first before I say the word. And then I said, now let me say it together. January. This is difficult for them because some will say three. Some will say five. Some will say four. And I'll say, let's try it again. Let's do it together. January. So we teach them the months of the year. Then let's do one syllable. May. And then August. Two syllables. And then this is another one. Saturday. And they love doing that exercise with us. It's a fun game to teach your friends syllables and to improve on the fanatical skills.


So, what we have done? We have looked at basically properties of various kinds of sets. We started looking at a set. The limits of sequences may not be inside. So, we looked at sets which are called closed sets. What are closed sets? We have the limits of sequences of elements of that set are also inside. So, that we called as a closed set. Then, we looked at a set may not be closed, but you can have something called the closure of a set. So, a is always subset of a closure and closure is the smallest closed set which includes a. That is interesting. I think I should say that something more about it. We proved a result that given a, a closure is the smallest closed set including, remember this we proved that. So, this was basically by proving that a closure is always a closed set, a is inside it and closure of closure is itself. So, that is a smallest. But now look at this given any set a, a given is contained in say a whole space. So, let us write say Rn. a may not be closed, but it is contained in Rn which is a closed set. The whole space is closed. So, let us collect together all sets C in Rn. C closed and C includes a. So, look at the collection of all closed sets in Rn which includes C. At least we have got one candidate. This set is non-empty because Rn is member of Rn belongs to you because Rn is closed and it includes a. Now, if I take the intersection of all sets C belonging to you, look at all the sets which are inside this collection and take the intersection. What can you say about this set? Intersection that will be a closed set because arbitrary intersection of closed sets is closed and each one of them includes a. So, a is subset of this and it is closed. My claim is this is smallest one because we have taken the intersection of all. So, in fact this is nothing but this is equal to this set is equal to a closure. So, this is another way of saying what is a closure. Look at the intersection of all closed sets which include it that will be smallest and that is closed. So, it has to be a closure. So, that is another way of defining a closure. You will find it somewhere. Now, can you say what is a interior? Given a set a, can a similar thing be said about a interior? Try to make a guess. Given any set a contained in Rn, what can you say about a interior? It is the largest. Interior is always an open thing inside but empty set is open. So, I should try to make it bigger. So, try if you like, try to prove it. A interior is the largest, you can deduce it from here also if you like, is the largest open subset of a. So, what we are saying, look at all open subsets which are inside a, take their union that must be a interior. So, various nice playing around with sets and definitions. So, I am leaving this also as an exercise because we do not really need it but it is good. It will help you to understand interior points and so on and open sets and so on. So, let us see what. Now, let us come to a very important concept. We started looking at sets whose limits may not be inside, limits of sequences but there can be sets which have no convergence of sequences at all. There could be sets which does not have any convergence sequence at all. But possibly there is a convergent subsequence of every sequence. There are no convergent sequences but every sequence has a convergent subsequence. There could be sets with that property. So, they turn out to be very important collection. So, we say this is such a set is called a compact set. A set is called compact. If every sequence, the sequence may not converge at all but it should have a subsequence converging where to a point inside the set. That is the condition. Are you understanding what I am saying? Yes, subsequence converges and the limit is inside that set. Look at the sequence minus 1 to the power n. That sequence is not convergent. Every sequence need not converge. It may not have any convergence subsequence at all also. Look at the sequence to n. Neither the sequence converges nor any subsequence converges. So, convergence of a sequence is independent of anything. But some sets have that property. There are sequences. Some of the sequences may converge. Some may not converge. Those who converge collect the limit points and put them together in a box. We call that a closure and so on. But now we are looking at those sets which have the property. Given any sequence, it should positively have a convergent subsequence, at least one. So, for example, let us look at examples. Look at convergent to a point inside the set. That is important. Keep that in mind. Look at the closed interval, say 0, 1. Take any sequence in that closed interval. Take any sequence in that closed interval. We proved that given any sequence, there must be a monotone increasing or monotone decreasing subsequence. Given any sequence in the closed boundary interval 0, 1, there is a subsequence of this sequence which is monotonically increasing or decreasing. It remains between 0 and 1. It is bounded. So, the closed boundary interval 0, 1 has the property that every sequence has a subsequence which is monotone and bounded. So, it must converge by completeness property. So, the closed bounded intervals have that property that we are looking at. Namely, every sequence has got a convergent subsequence converging to a point inside that set. So, any closed bounded interval a, b is a compact subset of the real line by this definition. So, I am giving examples now. So, example 0, 1 is compact. Is that okay? It is a compact set because given any sequence by our earlier theorems on sequences, it has a convergent monotone sequence and it remains between a and b. So, it is bounded. So, it must converge. So, every sequence has a convergent subsequence. Let us try to look at say a, b. Let us take any sequence inside. Let us try to copy the earlier proof where it goes wrong if at all or it works for this also. Take any sequence in the open a closed b. Then it has a monotone sequence, subsequence. It is bounded by a and b. So, it must converge. That is up to completeness. But the limit may or may not be inside open a and close b because earlier for the subsequence every term was between a and b. So, the limit could be equal to a or could be equal to b. But that remains inside that interval. So, the limit is inside here. The limit can become for example, a. If you look at the sequence a plus 1 over n, then that is going to converge to a. Any subsequence that also will converge to a because the sequence itself is convergent. But a is not inside the set. So, this is not a compact set because we have produced not compact. So, it looks like this closed bounded interval seems to be a prototype of compact sets. So, let us try to prove a result. We guess a result now that a set in R n is compact if and only if it is closed and bounded. So, we want to prove a theorem. So, a is compact if and only if a is closed and bounded. So, let us try to prove if we are able to prove then it is true if not then we have a contradiction. So, let us try to prove. So, let us say a is compact. So, this is given. So, let this is given to show a is closed and bounded. So, how do I prove a is closed definition? What is saying a is closed? If a sequence of elements of a converges somewhere then that must be inside the set a. So, let a n belongs to a, a n converges to a belonging to R. Then we want to show a belongs to small a, element a belongs to the set a. What is given to us? a is compact and the compactness is every sequence has a convergent subsequence converging in the set. So, a n is a sequence which is itself converging by the compactness property. It must have a subsequence which is converging but to a limit which is inside the set. But the sequence itself converges. So, every subsequence has to converge to the same limit and that is in a because it is closed. So, small a belongs to. So, let us write then by compactness as a subsequence. So, let us write a n k, a n k converging to something in a. It only says it has a convergent subsequence converging to something which is in a. But that something is a because a n converges to a and it is a subsequence because it is a subsequence implying that a belongs to a. So, we are using the fact compactness is there is a subsequence which is convergent. But the important thing is compactness is that the limit must be inside the set. On the other hand, sequence itself is convergent. So, every subsequence must converge to the same limit and the limit being a, a belongs to capital A. So, a is closed. So, next I should prove that a is bounded. If a is compact, it must be bounded. If not, if it is not bounded, what will happen? If it is not bounded, it is a subset of R n. When do you say a subset of R n is bounded? We are not defined it as such. For real line, we define bounded that it is between alpha and beta. When would you say a subset of R n is bounded? Because you can go any direction now. So, there is a ball which includes a. Is that a good enough definition? We will say a set is bounded if there is ball of some radius. You can take it center at origin or it does not matter which is inside. So, if you want very precise, you can say we will say set a is bounded if there is some radius R, say that the ball centered at origin of radius R includes the whole set a. Good enough. Is it okay? Let us take that as a definition of bounded. Now, I want to prove if a set is compact, it must be bounded. So, our definition of bounded is there is a ball which includes a. So, if it is not bounded, what will happen? Whole of a is not inside any ball. Whatever ball I take, there is something which is outside. So, what will be the distance of. So, if I take a ball of radius n, n natural number, then there is a point outside. So, that point outside, how much away it will be from origin, distance at least bigger than n. So, it is outside. It is a ball of radius. So, there is a ball of radius n at origin. So, this is n. If there is a point outside, what will be the distance of that point? It will be greater than n. So, saying that a is not bounded implies. So, I am saying this implies for every n, there exists some xn such that norm of xn is bigger than n. Is that okay? Just whatever we are discussing, we will just return it to you. So, can this sequence xn have a convergent sub-sequence? Because this is a sequence in A and A is given to be compact. So, by that compactness, it should imply this should have a convergent sub-sequence. But can this sequence xn have a convergent sub-sequence? It is going away and away. The points are going away and away from 0. You can say the sequence xn is unbounded. Norm of xn is bigger than n. So, as n becomes larger, actually this is becoming larger and larger. So, this does not have, this sequence cannot have a convergent sub-sequence. Is it okay? If you find visualizing in R n to be difficult, you can keep in mind your real line. You have got points on the line which are going away and away from 0. So, sequence is unbounded. Convergent implies that is a necessary condition. Every convergent sequence should be bounded. So, this is unbounded. So, cannot converge by that also if you like. So, saying this sequence cannot converge because it is unbounded. So, that means our assumption there is something outside. Every ball must be wrong. That means the whole of A must be inside one particular ball. So, implies A is bounded. So, what we have done? We have proved the theorem that a subset of R n is compact if and only if it is bounded and close. That a less than b that is closed interval that is compact, then it is closed. So, the whole space is not R or R n is not compact. Real line is not compact because it is close but it is not bounded. Same reason R n is not compact. What way it is useful? So, let us come to it is closed and bounded that we have proved.


My skin on the daily patients I'm lying and burning while bathing applying the lotion I'm smooth as a baby I'm way too emotional. Damn it. I hate it.  They all want to date me I'm the type to be romantic on a tinder day rather just talk think it lost in the dinner place Sting your letter minds over coffee and a topic not important only phones get ignored now I'm horny for a second age never late struggles to be a young hopeless romantic flooded with love like I wrote That's how tending love is a drug and I'm definitely an addict. I want a good Fatus. I say that you just gotta have it. I just would imagine my wife. He takes care of her body and obvious sign She enjoying her life to the max our minds on the same page our lives be the ocean be the same wave Our time getting spent to the old age. Okay. Okay next level now


Clemens comes in from Sweeney, it's fallen into the pack of Andy Cook, Cook still, feeds it through, Jamie Walkenow, minutes to go! Bradford City retake the lead in front of the Treblings, they've got huge three points, it's his first goal of the season, and it's on the same ground where he got his first goal of his Bradford City career last season. What a moment for Bradford City, can they hold on now? They lead 3-2 with four minutes to go. What a goal from Jamie Walkenow, he has been immense in the second, he broke and it was Andy Cook that came forward, cut inside, he tried a freddy to Bidane Oliver, Bidane Oliver got blocked, it then fell to Walker, running through, one-on-one with the goalkeeper, he slides it into the bottom corner.


Shoot down the temperature likely. So keep your distance. Yes, sir Let's stay focused watch your sectors There's the communications outpost Could be the missing troopers. Let's check it out Put us down on the bluff eyes in the sky. So let's go six. All right noble team spread out Watch the approach Structure point three four looks clear from this angle Distress beacons coming from just south of your commander. We're close Brought to that eyes peeled Fair the beacon make out any ID Negative, but it's military Why are we not seeing explosives residue noble three? Can you confirm any EX residue in the area? Well looks like there's nothing here, let's move on Smoke at the next structure boss circle west and check it out Selective we don't need a telegraph or presence


Hello, this is Nick from Laptop Media, and today we will show you how to open the Lenovo IdeaPad Slim 315 Gen 8. To open this device you have to undo 10 Philips head screws. After that, pry the bottom panel with a plastic pry tool starting from the back. Please remember, it would be of great help to us if you just hit the like button and subscribe to our channel. That would motivate us to make even more and better videos for you. Here we have a 47 watt hour battery. Before taking it out, detach the connector from the motherboard. Then, you have to unscrew the two Philips head screws that are holding the unit in place. The capacity is enough for 11 hours and 7 minutes of web browsing, or 8 hours and 20 minutes of video playback. You can find this device with 8 GB, 12 GB, or 16 GB of soldered DDR4, 3200 MHz memory. The 8 GB configuration works in a single channel mode. For storage, you can rely on a single M.2 slot, compatible with 2242 Gen 4 SSDs. Due to AMD platform limitations, the slot is downgraded to Gen 3 speeds. There is a thermal pad below the NVMe. The cooling solution is basic. It comprises one fan, one heat pipe, a single heat sink, and a heat spreader.