All right, it's time for Off the Press, and we have a very special guest of, well, our Monday special. Punabha and Kateria, political affairs analysts. Joining us from River State, always a delight to have a Punabha. I'm speechless. Good morning, Punabha. I just love how you look. Perhaps that's why I was having a bit of a problem introducing it, but I just didn't know how to start introducing this movie. Don't worry, I'll be with you. Oh, that would be lovely. You're looking like Tartif this morning. Yes. Okay, so we'll start with the Guardian newspaper. Niger Invasion. Nigerian main care, over 2.98 trillion Naira, double defense spending yearly. Sanctions against Niger. So you have the picture there of Tenubu, present Tenubu, and someone in military uniform. I don't know if they're trying to show Wagner group there. So what's your take on all of these that's been unfolding since Tuesday when Mohammed Bazaam was removed from office? Yes. So it's quite interesting, you know, they say you don't have to admit and regret. We have coups every couple of years in this country. But these are, our coups are quite different. These are just true. These are coups that cover the niceties and complexities. We call them democratic coups. What is a coup? A coup simply means the foisting of an illegitimate government, illegal and illegitimate government. That's what a coup is. In fact, the military coup, I prefer the military coup to civilian coup. Because in the case of a civilian coup, what we call democracy. The truth is, you have falsification of results and all kinds of crime. In which you couldn't pretend to be a part of the government. They take over. It's as simple as that. In your own country, you've never been able to admit your crimes. You're going to move the law in another place. What's your business? Was it not the same country?  We also don't have right to tell Nigerians. The Nigerians are completely happy with that. You know, I don't believe in that philosophy of the worst democratic regime is better than the best military regime. This is complete philosophy. Till tomorrow in Iraq, they are regretting. Till tomorrow in Saddam Hussein town, they are regretting. They would have allowed those so-called leaders to move. Today, they are suffering. Just like in Nigeria. We are suffering. If we have never had it, it's bad. Not even another thing. That's the truth about it. Your life is all about living. Your only essence is how you live. If you're living comfortably or not, we cry against the new regime because we saw the media have been drunken here and they're depreciating. But what is the situation of today in Nigeria? Is it better? Can you even compare the batches you live to the new regime? Do you have enough for them to post in Nigeria? Can you even compare? Even they are so-called demonizer batches. So, we don't have no business. Now, we are going to spend too much. We are not going to be able to address the burning and nagging issues in this country. People are dying on daily basis. People cannot fend for themselves, for their families. Things are getting worse by the day. Economically, politically, social. You are not going to address what is going on in this country. You want to go and waste resources. Why are they doing it? Not because they believe in democracy. Because they are equally not democratic. It is simply because they want to protect their own. We could just stop this. We say, how did you and these people get overtaken by this? We could not stop this. We are going to be victims. Now, this is supposed to happen. The next thing will be Nigeria. We are going to spend our hard-earned resources in the name of trying to protect the local. Who sent you there? Who sent you there?  Your house is on fire. But we are big brothers. We are big brothers. No matter what. Since we are big brothers, we have to go and do big brother affairs even if our house is on fire. So you save the younger one. Isn't that what is happening? Isn't that insanity?  Your kids are dying. And somebody else's house is not on fire. They are just talking about changing your government. And somebody else's house is broken. You need your own program to put out the fire in another position. Isn't that a stupidity? But for this international revolution that is in place and so on, did you see how elected you on the streets? With the previous government, the Democratic government, the so-called Democratic government was also autocratic. And they were not pleased with the style of the government. That's the meaning. If you happen in Nigeria, it's not fair if you are not there. Another thing I do not like... Open up. Open up. Open up. Another thing playing out that I do not understand and I do not appreciate is the fact that this is an aqueous thing, supposedly supposed to be an aqueous thing, but it's beginning to look like it's a Nigerian versus Niger situation. Tunubu President, Tunubu just the chair of aqueous and he is speaking for aqueous and not for Nigerian. Nigeria is not at war with all looking to go to war with Niger. So why is it playing out this way? I'll tell you. In the last aqueous attacks and how it operations, tell me who has been something largely in terms of personnel and finance in Nigeria. The aqueous community will be shouted. But when the teams are down, they'll leave everything to Nigerian. And because of sheer man pride, neither at that point in time will not be able to leave them. That's the truth. Check all our aqueomog operations once sought by Nigeria solely in order to just bring some pinnots. Last chunk in terms of personnel, in terms of finance will come from Nigeria. And that's where Nigeria is at war. If you doubt this, wait and see. If at all they are going to commence this operation, wait and see. Nigeria. And that's why they say no. We have enough problems in the house. Solve our problems. Solve our problems. Forget about the international. The same Nigeria will say we have to do the same Nigeria that will do something. It doesn't matter. It's a question of nomenclation. Democratic regime. But the question is, can't you see the Nigerians are so happy? Can you love the Niger, more than Nigerians? The Nigerians are so happy. Just like the day at Batcha died. That was the same expression we saw on the streets. The Nigerians are so happy that Batcha died.  But because of the frustration in the land, but when you compare the Batchas, you will not compare. So when you have a government, an autocratic, selfish, heartless, scarless government in power, and you say the people should just relax, they should not talk. They are not doing it because of the Niger, because they love the Niger. They are doing it because they are scared. Because look at the Sahel here. People are just taking over. And they are scared that one day it will be mine. They say, where you killed the goats? The sheep is not happy. Tomorrow it will be mine. So we better stop this school. The only way to stop it, anywhere in the world, is to ensure good governance. That's the only way. So in consciousness, power needs power, less concern. The former lasts first, but the latter lasts last. So it is not by mice. Not by mice. It has to be good governance. Give us good governance. You will not have food. Give us bad governance. You will have food. It's a matter of time. And that's what they are scared of. That's what they are scared of. They don't waste our resources. We want to go and protect democracy in another country. The same story. The same story is on Daily Trust. Nijek who is adamant as deadline ends. Now we know that they have closed their borders to Nigeria. They have refused access for our soldiers to enter Nijek even though there was supposed to be a joint force that will fight insurgency and all that. They are on their own and they are refusing our people entering there. The troops deny Nigerian soldiers access to Boso, Ekoaz, Mum over next line of action and the president has met with the governors of the seven northern states. What do you think will actually be the discussion like? Because they are really silent about what they talked about in that meeting. But what do you think that discussion will be like? Pretending to be walking. Maybe nothing. Let's see if they are also... What do the governors say? The directives come from the CC. Here you have the military. Let me tell you, they are scared. Nigerians have been emboldened by countries like Russia, foreign countries that will just drop one Nigerian and that will be the end of this. They just try to play the big brother role that you are not. You are not. You are not. A big brother that cannot put a house in order. Is that a big brother? They call the big brother. They get a deadline. What happened? They just got dumped at all. No house of fear. A big brother. He did not challenge Nigeria. He did not challenge last. I am challenging you to do your worst. And I bet you know how to come up with a community or less Nigeria without foolishness. Who live in death. Because if you look at those behind you, there are a lot of foreign countries behind you. And they are waiting. They are not there. Nigerians have to pay their taxes. So my dear, meeting with the several northern governors is completely telling them to ensure foreigners and Nigerians don't come. But this is every time we have Nigerians moving in and out of the country. In fact, they are more prepared than us. Because you find them moving into kill farmers, headsmen and so on and they go back to the community. They know how to rush the country. So the people are not scared about it. No house of fear in them. So how do things with one billion governors? And the governor sees you are still the city that will give the order. God sent you now the borders of the governor. So having me to the refugee, having me to the governor, farmers, headsmen, classes and stuff in fact is hiding on daily basis. They have been having me to go to the churches in the north to kill people. So what is the need of what we need? That's what we are telling you. If you don't put your house in order, you are going to DJ. If you don't put your house in order, you will not be able to save your child. You are going to the next house to save the next master. Because what do you want to prove? A superhero. Your own child, you will not be able to feed your own child at all. You are going to the next house to feed another master. What do you call that? What do you call that? So apart from this year, you have Mali, you have Sudan, Chad, and then you have Burkina Faso. I don't say the Sahel. Yes, and so you have Burkina Faso. The guy in Burkina Faso, it's appearing to be the new kid on the block. Everyone seems to be liking him. Did you listen to the speech he made in Russia recently when they had that summit with Russia? Africa Russia Summit. And what did you make of the whole thing? Oh, you did it? I did it, but do you know how you pronounce the country's name? I have a strong belief you have a sign there. I am a broadcaster. I am a broadcaster. I am a broadcaster. I am a broadcaster. I am trying my best to pronounce this word. I know. I know. So these things are unfolding. These things are unfolding. The military boys are becoming popular on the continent as we speak. If you go across social media what you see is that these military boys are being liked by Africans. Why? Because a lot of the things that have been going on especially with regards to France and the West and all that's been playing out with the way African natural resources have been plundered by Africans being left in poverty. Almost in conniving with our so-called leaders it's beginning to make Africans across the world say wake up Africa, let's take our destiny in our hands and it does appear that people are beginning to think that it will take or require the involvement of the military boys to be able to achieve this. Is that an accurate analysis do you think? So after analysis you know because the citizens have lashed into a state of victory where the microcosm to you that I charge of the resources use the resources to oppress so you can't talk. They buy the judiciary and that's why I'm always worried when the judicial is compromised. That's why even in law once you have you keep doing the sincerity of the job that job will definitely be the same. You know I always talk of justice he was justice he was the one who after me then we are not going to be seen to be done but we are not going to be seen to be manifested now. If you look at the circumstances just because a tactic followed them into the chambers and came out and just turned to summarizing and they are practically why did you ever move in? And a few people said yes whether you actually discuss with them or not the same accident is perceptual as you gone into compromise in the process therefore to set aside the time and that's why the judicial is key look when you have these foods and you see people come out endostical it's a referendum of the people's bad governance me is democratic me is military it doesn't matter the nomenclature does not put food on your table what puts food on your table are the good policies from the good government and you don't voice candidates on paper if you are not elected and eventually you'll voice on the Nigerians or Nigerians or the people from any country what's the difference between you and the Coupes what's the difference because the military man that takes over is not somebody in fact the coup is even back the coup is even back in that case you are not even aware there was no selection you just came this one you chose this is the man I want to be my governor, my president every spirit you voice on tyranny the tyranny is not just keeping the man down giving me somebody I did not vote for it's tyranny to hell what can we do to hell what can we do I will be there I will do something I have little ideas if you step on it you will blame so don't take anything for granted I don't want to be in trouble our side open up do you share the concerns in some quarters that Vladimir Putin and NATO may end up settling their scores on African soil and using this whole thing that's playing out as an opportunity to come and settle their scores here and we know what we are talking about weapons we are talking about weapons it's it's what do you think that is we are hearing okay that does is name you see what is going on it's like a woman who is buried now chooses her the second wife on the fourth wife or a student who is still not intelligent will come back and tell the theorist because he wanted to sleep at night that is the only you know that that is not the case why do you create Africa why do you create Africa so if you are a leader and you fall to such shenanigans if as you mean without considering that means you are not what we call a leader that's the truth because if my child comes to me to say that then we are petitioners first granted because I look at the script so if you are a leader and you fall to such a shaman shenanigans that they wonder why are you a leader first of all that disqualifies you to me sorry do your best if you have a country if you are the president of a country and you are loved by your people your people will protect you you don't even need soldiers you know what I might say so at least of reason and purpose of life that's the point if you have done that look at that they will say that they have to evade and funny enough they sold all kinds of dummies and all kinds of stories against it today the people are angry the people are bitter we have gone past that stage that was not even the media the social media era now we have the social media era the world has been more enlightened the previous man or the American man or the Russian cannot trust them and talk to you now and you swallow everyone and see them we did all that when there was an inferiority complex now we know what to do they are talking of what the equipment and so on they are doing that but he will do that the equipment they use although some cases also most of them were manufactured in the east there what happened they killed this although we have been one of the superpowers they do not say about Russia there is this policy Russia is not like that most true poverty I go to what makes you a leader so I don't believe it is just theories the question is ok we agree that they want to use this as a fertile land to test before that time how have you been governing the people because the people would have come out to protect you rather they are coming out to condemn you and then dusting people what does that tell you what does that tell you so you are not going to dust your own people and dusting your people against you that tells you how bad a leader you are ok let's take something else I'm just not happy that we wasted our money building a rail line to Niger now they are catching us up from their country because of something because of something that could have been said we should have been brothers and sisters but now see what is happening but let's take another topic because the former president is from Nigeria recently he said his brothers and sisters are there so it is from Nigeria he did say that anyway something happened something happened at the weekend something happened a few days ago a doctor lost her life if you go to the punch news paper you will find that story the doctor lost her life in a lift I don't know if you saw that story and now the governor has suspended the agency's chief the agency that was supposed to be the president's and he has also the facility manager and because of the death of this doctor we see the story surrounding what happened she was from the 10th floor enter the lift the lift failed and crashed down one hour later they were able to remove her and stories surrounding that incident say that everything that was supposed to be needed to give her the life that she needed to resuscitate her was not there like the blood oxygen and so many other things even though the government has denied this so now sacking of the facility manager and the suspension of the agency chief do we need to start applauding the governor or you think there are some other things that should have been done in the aftermath of this unfortunate event first my heart goes out to the family I mean it's so, so, so heart wrenching that you train a doctor a doctor a child to that level only to be killed because she was killed was killed by the managers of that facility and so it's criminal negligence it is not enough to sack me they should be prosecuted and jail they killed her it was just not the talking of meter you can kill the brand of the gun with a knife even little before they killed that doctor it's criminal negligence so it's not enough to sack one of us they have to be prosecuted if they just sack I believe that the family members will go ahead and even if the governor looks at that then he knows it's not even paying them for the facility if it's a governor because I can pretend we just sack and not to suffice the medical doctor what's the difference she's gone, she's gone her siblings, her parents do you know the cycle of they could lead to the death of the pregnancy and that so you ask the question if it's enough, no they have to be prosecuted and jailed there's no excuse they will not be killed but they will be jailed and if you also investigate the relationship between that facility at first, even if you give them money to be taken that's not an excuse I'm not saying that's an excuse because it's in most times we run into such problems as a result of the inefficiency of the government forget the lies forget all those rebuttals they lost a lot of answers they don't say about three years after what happened I don't believe what government officials say I don't okay, I will not mention it you have someone who said I will never accept this I am bigger than that I will not do this they are the person who tends to accept what is less than that so the people who have the government devoid of right to do it so I am not bothered about the government's rebuttals I am not bothered about the government's defence I am not interested at all with the government's I want to see that this matter is engulfed and dealt with the doctors are taking it off and the infinity because we are going to cut and remain there for twenty years that's why I am worried about the kind of political judges we have the debate the judiciary is in a mess it's a lie because it's supposed to be the best year because if you complete it you don't have to book a shower once it's wife the president will come to appeal I think you don't really understand what it means when the judiciary is compromised and better the judiciary can be compromised for the judiciary to be compromised then that nation is doomed that nation is doomed because once the church sets in the nation is on fire once you don't believe even justice is denied you cannot go to a government to do that after that mess they go to courts and they use the court as a cover and why the president who could bring people with him you wouldn't understand unless you are directly accepted you will not understand so when it goes to that of the lady on Sunday she is gone she is gone there is nothing you can do about that she doesn't even feel any pain anymore just before she died what of the others there has to be punitive measures that will serve as deterrence you could have been me you could have been you she was going outside to take and put the others that was what we were told so she wasn't even hungry she died hungry and mom moves to tears because of the negative of the government and you come to tell me you suck I know suck program it as the system well the medical association they have taken this up seriously and they are digging and investigating and they are making demands and so this I know is not one case that is going to be swept under the carpet the facility manager has been sucked but a lot of people will need to be grilled over this matter because sometimes facility managers are not even enough resources to manage the facilities yeah there is a lot of politics going on that's what I said they should have sealed off the facility if it was that bad probably they would have written memos for memos or somebody collected the money and just even those who installed the lift who knows the quality of lift that was installed and there is so much monopoly in that sector no matter how bad the person who installed the lift if he discovered that this person who installed the lift is not reliable you can't even get another lift operator or lift servicing company to come and service it you are mandated to continue to use his services so you can sabotage it except to remove that lift just imagine if you are on that lift going to see somebody it could have been you except you removed the lift and made another company to import a new lift for you your son or daughter was also on that lift going to see somebody that would be very painful even if it's a cost to you, you are gone you don't feel the pain anymore but imagine your son or daughter was on that lift how would they have felt you wouldn't even be on that lift this morning you would be so traumatized that you wouldn't be on that lift and God will help you to come back to that lift it is a very sad incident I am really paying you unless you've never trained anybody before you know what it takes the mental stress and so on every morning you know this is the bad thing and you are happy and when the child is doing well the parents of her family what is going on I am not wait for investigation people will have to go I don't know how let's still just say this as we round off this segment let's just remain with the hospitals as it is or the medical doctors doctors are on strike and because of that I know one or two people that are directly related to me that have passed on because they went to hospitals that doctors couldn't attend to them and those were the nearest doctors or hospitals that they had but government hospitals and of course they were not there but now since the doctors are on strike the federal government is still talking about no work no pay as the only solution to scare them back into work and I am sure the NLC is also spoiling for a fight because they were talking about litigation against them and all that, no work no pay all the time using this to frighten the workers to do what they are beating at so what do you see about this as we wrap up on the segment please doctor strike no work no pay the NLC let me talk about the NLC the NLC forget the NLC the NLC is a great disappointment I must tell you this whenever they don't have the money to feed in the morning or buy their fuel they threaten the strike action or when they want to see the face of a president they threaten the strike action then they are working, see that's why the president cannot let them rubbish if you are backing on the strike you are backing on the strike the demands are met, you don't call off on this they becoming on strikes you mobilize people to go the rest you call on strike as the government ever, ever to fill any condition the question is no so why do you keep calling on backing on strikes and calling on strike they can't see you they can't see what they can get from the government whenever they are working they see the president they see that's why we've met nobody should listen to the NLC anymore there was any NLC who had answers the NLC has lost its value complete absolute rubbish nobody should bother about the NLC those are just people they see the office has a one gravy tray how to fed off like their pockets that's all going to the doctor to go there on time so I don't have to talk about a bonding one as NLC there are bunch of dissent why are you calling on the strike your demands have not been met you say you agree to accommodate the next week the government behind you will go to court the government they employ what they call the fabian police they are negotiating what you may have or what you have and they are working on something that's the fabian police and the NLC need that and they tell them that they are not sees of these facts they are going to collect any payload that's why they see that and they know nobody should bother about NLC bunch of dissappointment the NLC they go to court and go and go on these are just people who want money the NLC would really have somebody paying that same they have gone let's just go to the doctors no work no pay no problem you know it's difficult it's hard but i will opt for no problem no work no pay by the time like you said some of your relatives have died by the time the hospitals are littered with deaths that human being that is so colored to say no work no pay when his own wife or kids are killed relatives are also are also killed by doctors by doctors when there is a resolution i will see if it will be a life to say it again no work no pay you see the truth is something has to happen in this country something has to happen in the i will say what we have is socialism for the rich and capitalism for the poor to hell for the economy the leaders are leaching on Nigeria something must happen i'm saying this must happen if otherwise something must happen if it happens the wind is blowing and i pray it comes this way the wind will change good that you said and i pray there is this thing they say pray until something happens there is something that is going to happen let's hope that it will come from God about this is where we will have to draw the cotton everybody is created by God we will have to draw the cotton on this segment here today thank you so much for being a part of our program always it's a delight to have you with us thank you good morning so we will take a short break and return in a moment stay with us


Good afternoon church family, Pastor David, welcome to Unfiltered, a time where Pastor David is unfiltered about what we share just for an event. So thank you for tuning in and look forward for you joining us. Today, Pastor, I wanted to ask a question more, what is the thinking behind this, I don't know what you call it, this move where Pride Month is celebrated for an entire month. But we look at Memorial Day, it's one day, Independence Day, one day, Veterans Day, one day. And these are our significant things that men had laid down their lives for, our freedom. They were able to, we celebrate some very significant things on these days. And so now we hear of LBGTQ celebrating a whole entire month. And really, what have they really done for our society? So what is the thinking behind that, Pastor? And it's demonic. I mean, the simple answer and the quick one is simply demonic. It's a demonic spirit that is leading people to accept and to actually normalize perverted lifestyles and perverted desires. And this isn't new. This is something that has been under attack in terms of the family, in terms of morality and things of that nature that's from almost the beginning, John. Yeah, I asked myself, what is the reasoning behind making such attempts to glorify such a perverted lifestyle? And like you said, veterans, those who serve the nation, willing to do whatever was necessary to preserve our freedoms, you get a day where people may or may not appreciate the sacrifice that many, including myself, made to at least be ready to, and many others did, defend this nation and to the giving of their own life for it. That's why we have Memorial Days a day. And so you have Veterans Day, Memorial Day, and maybe one or two single day events.  And you have to ask yourself, just what is it that they do that they're so proud of? What is it? What makes their choices in terms of their physical attractions and behaviors? What makes that something to be proud of? I don't see a heterosexual month. I do see attacks on heterosexuality, but I don't see a month of it. But that's what it is. Pride, even saying it is something that you celebrate pride, again, pride for what?   Is that what you're saying?  I mean, that makes absolutely no sense. And so what it is, is an undermining of the traditional structure of the society.  I would like to see if they're in China if they have a pride month. I'd like to see whether they have one of those in Russia. Do you have a pride month?  Do you give them high positions in government? Is that what you do? Do they have pride month in Saudi Arabia?  And what it is, it is a demonic spirit to undermine the foundations and the traditions of the family in the United States. And it ought to be opposed. It ought to be at least spoken of even as we're doing right now in the sense of saying, what's the spirit behind it? The destruction of the family. That's the spirit behind it. Who is the one initiating that? Well, it's demonic. And we have our administration applauding it and lauding them heroes and brave where we don't even really get that on the days that we're to celebrate. That's really significant. Well, that's ridiculous. You know what? And I know what, John, I don't think that a person is to be lauded based on a sexual preference and activity. You ought to be lauded for the accomplishments that you made.  That's wonderful. You did something wonderful. And I truly appreciate you. But what has happened is people don't want to talk about what they do, the kinds of things that they do, the kind of behaviors that they have, the percentage of those who are actually involved in those kinds of things, because they're exaggerated. All those numbers are exaggerated. I mean, the Kinsey report years ago said, I don't know, 10%. But in fact, I was looking at a study recently by a particular department in UCLA that said that the same sex percentage is somewhere around between one and 2% of people in the nation. And you said, oh, no, it's just filling it. And the transgender is less than 1% in our nation.  They're nude. He can walk in and call himself a woman. He doesn't have to have any appearances of transformation at all. And you can't keep him from doing that.  No, this is an attack. And it's an evil that's coming down from our present administration. But it's not just in that administration.  But you see the corruption of that. So this is just one evidence of it. So the Dodgers caved. They caved to a small but loud group. That's what they did. And they ought to be ashamed of themselves. And I'm blessed to know that at least one Dodger stood up and said, I'm a Christian and I don't go for this. And the other that everybody knows who's I've admired for a long time, he seemed to have no spine. I don't understand that whatsoever. I'm very disappointed in in Kershaw who happens to be somebody that I greatly have appreciated up to this point. But to back down, no, you don't. There are things you draw a line and say, listen, if you want to celebrate this, I'm not going to judge you for what you do because I don't. But to try and make me agree with it and say that it's right and it's good, I will not do that and can't pay me to do that. And so believers are supposed to be salt and light. So for us to yield to that, for whatever reason, with the Dodger organization, for what? They're losing fans. They're losing fans. It's ridiculous. And they need to stop this nonsense, especially bringing in those those perverted sisters of perpetual indulgence and to give them a hero award. It's sickening. And I wrote to the Dodgers, they don't care, but I wrote and I told them, this is just wrong. And as I've been a Dodger fan, if you will, since 1959. I mean, and for them to give in and cave in in this, and these other, you know, and the angels to celebrate it, the angels fallen angels.  But yeah, that's what is the reason they do this, John? It's because it's a demonic spirit to destroy the foundations of the nation. And if the foundations are destroyed, what can the people do? And as Christians were to stand up and yeah, we don't have to be mad. You know, it is something that I have emotion over it. But I'm certainly I'm not going to attack somebody. Listen, I've known more than one homosexual, including my sister, who came out of lesbianism. I've known more and loved more than than, you know, just one. But the but it's still true. It's what they're doing is wrong. It's going to bring judgment.   And this other one had them all over the place. And they've done nothing. And they'll do nothing for the son either is his son who made millions of dollars you know, working in a field he has no experience in and nobody notices that they're so busy hating the orange man. And I'm no fan of trunk by any means. I think he was a great president. But he the way he acted was really bothersome to me. I didn't appreciate the way he he acted and all and he's even acting now. But what's fair for one is fair for the other. And these this this is so obvious what's going on. So it's all part of I think a spiritual attempt and it's succeeding to undermine the foundations of the United States. And so pride month and this and that, you know, it's the elephant got its nose under the tent. And the whole body is now underneath it bringing in every other perversion that follows that kind of sexual deviation. And it's attacking our children. It's destroying our children. I mean, you've got children who think that they are something else in laws that are being passed or at least being a here in California that says that I as a parent don't have the right to tell my child that their idea that they may not be a male or a female that I don't have that right anymore. But Big Brother does and foster parents have to sign agreements that they're going to allow a child who is seven years old who can't even figure out how to spell his last name is able to tell you what gender is just because he thinks because he plays with it all. This is crazy. We're seeing it happen. And we have to be more vocal about it, John. Standing truth on God's way. Of course. I mean, that's what we're supposed to do, right? Well, Pastor, thank you so much. I wanted to take it to me just backwards, you know, and and but again, it's the times we're living in and and to pray for our nation, pray for that people would be delivered and set free from a lifestyle of sin and pray that our church stands up, not just our church, but the church, the body of Christ stands up. So, Pastor, thank you so much. Thank you guys for tuning in. Do want to remind you that we have our Wednesday evening services.  And it's a great opportunity to come be refreshed in God's Word.  And to be refreshed in God's Word, be in worship, and be in fellowship, it's a great time just to get together with with our church family. So I want to encourage you guys to come out 7 p.m. on Wednesday and then again on Sundays at 8.30 and 10.45. Tomorrow, it's Romans. And then Sunday, we're still in... We're going to be in Acts chapter 2. Oh, cool. Cool. The church. It's probably time to stop talking. Amen. Thank you guys for tuning in. God bless you.


In this video, we're going to apply the remainder estimation theorem, otherwise known as the Lagrange error bound, to find a bound on the error associated with using the sixth degree Taylor polynomials centered at x equals 0 to approximate the function cosine of x on the interval negative 3 fourths to 3 fourths. Now, according to the theorem, we need to identify a value of m such that it is the maximum value of the seventh derivative of cosine of x on the interval of interest. Now, the seventh derivative of f is sine of x. So let's take a look at the graph of sine of x on the interval negative 3 fourths to 3 fourths. We see that the graph is strictly increasing on that interval. So the maximum value of the absolute value of the seventh derivative of cosine of x on the interval negative 3 fourths to 3 fourths is sine of 3 fourths, which is less than 0.682. So in the theorem, let's let m equal 0.682. So according to the theorem, we have the error bound is bounded above by 0.682 divided by 7 factorial times x minus 0 to the seventh. This 0 is because that's the value we generated the Taylor polynomial about. So the sixth degree Taylor polynomial approximation of cosine of x centered about x equals 0 on the interval from negative 3 fourths to 3 fourths has an error of at most 0.682 over 7 factorial times 3 fourths minus 0 to the seventh, which is less than 0.000181. And this is quite good.


everybody forced here being the immaculate luminance of the elite patches six box NFL mixture number five random teams let's go ahead and do our random and then we will rip who luck again it's two plates two luminance one elite and one iMac legion all 2023 go six times names first one two three four five and six got kct down to diancy and then six times on the teams one two three four five and six uh jags steelers down to the cardinals luol dang is in chat now i've got kct jags steelers bret oh the chiefs matthewc rams tyson with the browns becky j giants matthewc niners ronald a packers nick embroncos kyle h cults um nick m pats jets brandon l raiders chat h falcons berry y with bill's cowboys dian has the titans berry y bangles buck star buck lions keith b texans kct chargers swaggy with the saints becky j dolphins commanders edward g vikings erik c panthers lin m seahawks zachery c ravens brian f bears kyle h eagles and dian with the cardinals all right results are posted to the site here comes your link find those anytime elite sports cards dot com let's do it all right start off with an orange rookie jetevis marten for washington to 399 here's all right they got eight um aiden o'connell michael mayer terry wilson number 999 rookie is j year brown for san francisco and a penthouse rookie auto cowboys douce vaughn for hoping for a big i don't think there's any luol dang autos in here i would jager mayfield's pretty excited about it got mcaffery to 77 for the niners yes all that andy got a darien davis rookie number to 999 for the chargers etched in time autograph to 149 is levon kirkland for the steelers guys smith and jigba on the rookie elitist for seattle rookie sambel port it's a 99 for the lions nice color match on the blue you'd say they're subred picket penthouse alliteration there you get ricky to 999 kamari connor for the chiefs patch card buffalos josh allen 31 to 49 four colors oh geez it's kyle biz mag beyond bow yeah it is got tidal waves of 25 is patch mom's shawnt can't take it anymore biz mag is actually kyle for sure come on kyle that's why he's modded look at that modded out of here yep all right man 399 is josh allen for the bills yeah why are the why are the biggest trolls of late in sports cards employees of late in sports it's ridiculous man embarrassing got silmin ricky to 149 for care uh for cleveland probably have a group chat laughing so hard that is a field vision of hurts the 349 eagles and number to 10 is a jailing carter eagles congrats on the jailing carter throwback design there all right two luminance now everybody sold out tonight kyle's live at 10 am eastern alphanatics live contenders optic basketball bowman chrome baseball mosaic football and flawless football jigba that is kashan bute number 250 rookie for the patriots a savage of acres and redemption joey portage junior year one signature stealers anderson there is a and uh alvin kamara number to 250 for the saints plate and tune cardinals rookie auto to two to start at 350 on the tune engage nick bosa allery downs witherspoon there is a um davis allen to 200 rookie for the rams there devin mccordy patriots superbowl auto unforgettable tyrie hill dolphin stink dude tyrie hill's not good right young rookie dual patch ravens or dual jersey roquan smith at ray lewis reflected in materials there's a gibbs lions rookie 18 of 75 nice jamir gibbs got a shot of watts and 100 for cleveland's and savage of etn to 100 for the jags i'll have to see what two of us has on the comeback in chat oh yeah that's right you went to kyle's you know you were jake lives no but i'll find it it's a big stinky house i was gonna say you could probably get it like we could probably find that on bamboo hr or something like that pretty easy got to 100 rikki stetson bennett for the rams zay flowers rookie auto to 50 boltamore and kashan butte full of a stench that bad huh he's in here ties it up dude double steal i want to move man dude got the third stand up is that carol i guess it won't no butte there's a rookie to 50 is adison from minnesota and an eric gray giants rookie auto to 350 and anthony richardson on the vertical rice young two uh number to 250 for the dolphins got a macintosh rookie auto to 350 for seattle fusion there is a jesson jefferson no not currently chad isn't broken in a little over a year or about a year now i guess he just stepped away he was still really want to break anymore bjohn robinson rookie but he's walking back anytime left on good terms got parson's jersey for the cowboys and hurts number to 150 for the eagles richardson colts rookie to 200 nice richardson and vestige of gara wilson all right let's do plates and patches now two of those got a base champ bailey it's all good chad it's good to see you back buddy yeah i think it was about a year ago brad just stepped away just got burnt out i guess shoals of 35 taxes everybody oh i i still talk to him though i still i still text him every week or so at least 35 taxes dull the shoals he's still breaking brad that's for sure zay flower steel signatures rookie auto for the ravens 10 out of 10 great hit ravens good break for the ravens too j caner rookie auto for the saints that one's number 250 flawless oh yeah football and basketball triple patch number to 99 is russ for the broncos yeah a lot of zay tonight offensive last couple boxes here he's big god deon he's like 24 to god deon he's definitely i don't need to be a shortstop for everyone like second baser or something got deon there oh that's nice uh deuce von number to 25 he's been all right brad the broncos just stink russ has been like okay much better than last year jamir gives rookie auto to 149 for the lions i'm with you there he's been like amazing but even if he was amazing i don't think it matters with that team right now nice gibbs auto 149 a schoonmaker cowboys rookie auto out of 300 and dual patch bills rookie dalton kakata 25 keep trying to do this all right i'm actually just maybe for false and in terms of what is going on stuck i don't know what to be stuck to what it was so this is good because both teams need to win this game not to 99 you got burrow for the bangles jerseys zack evans to to 99 zack evans i think is the rams yep rams next is kenny macintosh jersey auto for the seahawks to 99 it's not a bag topic chat it's definitely not a bag topic oh number two uh 49 it's edger tillman for the browns like i said man it's all good brad's happy so yeah that's what's up dude plus we're doing we're doing well he's doing well i don't see the issue he's still in touch with obviously rish's everything rish's his uncle they see each other a good amount that still lives around jersey auto is charbonne for the seahawks as well to 99 yeah people come in and ask him about brad all the time as you can imagine he was like the main breaker of like five years and then to 49 is a jackson smith and jigba auto for seattle seattle's at three they had the macintosh to charbonne and the smith and jigba dude seattle crushed the iMac collegiate box there good stuff one moment we'll do a quick recap then we're on to flawless did i oh man that feels like forever ago that was only like three years ago though yeah because it's almost october 20 yeah that was like three years ago jesus it's forever but it's also not that long ago relics of zack avans got kinkade russ mica uh roquan and ray lewis josh alan autos of smith and jigba charbonne sedrick tilman kenny macintosh luke schoonmaker jamir gibbs jake heiner ze flowers at 10 kenny macintosh eric gray ze flowers debon mccordy a clayton tune redemption joy porter jr steelers that was year one from luminance levan kirkland and deuce von all right y'all we're on the flawless now thank you very much


Tonight our stage will be taken over by Inner Sanctum and the huge wolf. We'll get a chocolate too and an excellent mystery story from Inner Sanctum Coast. Great. We're at your door, Mr. Holt. So open it, won't you? Welcome. Come right in. There's room enough for everybody. As a matter of fact, the more bodies, the better. Don't be surprised by the gloom at our new paint job in black so the blood stains won't show. And now, friend, draw up your chairs, dim the lights, and listen to a story designed to free the spine and set the teeth on it. It was written by Michael Sklar and Richard Manoff. And it's called The Confession. Now a star is Santa Sotega who plays the role of Alex. The last thing I remembered was the gunshot. The thought of the bullet. And the burning pain in my stomach. And everything got black. When I regained consciousness, my impression was of blinding whiteness. Don't try to move, Mr. Sturgess. Where am I? This is the emergency ward at the general hospital. I felt no pain. Just an overall numbness. And a desire to sleep. Am I going to die? Yes. How much time have I got? A couple of hours. Doctor, I... I want to make confession. I want to confess to... murder. I told the doctor about the money. And I know she was much younger than I. And she thought because I owned a little drugstore I must be prosperous. And I suppose that's why she married me in spite of the difference in our ages. But there wasn't any money for fun, or nice clothes. So she became restless. But I loved her. And I did my best. Until the other night, in the drugstore... Overclocked, Nathan. Another customer might come along. Just like an old man, chasing pennies. Never getting anything out of life. Reasonable, dear. I'm trying to make a living for it. Is this your idea of a living? Is this what you promised me before we were married? I know it's not what you deserve, Lenore, but someday...  I must have been crazy when I married you. So, no. I'm going to say something I've had on my mind for a long time, Alex.  I don't have to put up with you. I'm still young, Alex. Out in the open. An old man with money would have been acceptable. But because I had no money, he was going to leave me. I wanted to plead with her. I would have gotten down on my knees. But before I could say anything, the street door... And the man stumbled into the store. I helped the man to a chair.  His breathing came hard. Bullet, my chest, patch me up. I'm only a pharmacist. Do you need a doctor? I'll call Dr. John. No, I don't want any doctor. I want you to help me. Is it because doctors must report bullet wounds? Is that why you want a doctor? My reason. I've got plenty of do. I'll give you $500 to take care of me. I'd better call the police. No, lady, no. Look, mister, take care of me. Give it quite a... I'll give you a thousand. A thousand? Let me see the money. Here, it's a thousand dollar bill, but I've got plenty more. Lenore, help me take him into the back room. Oh, but Alex... No, I'll take him myself. You get busy and wipe the blood off the floor. Oh, no! No, you can't! Wipe up the blood. Do as I say, Lenore. Yes, hello? I'm looking for a certain medicine among these bottles. I mean, something... something... I'll be ready for you soon. Lenore, did you get the floor clean? Yes, it's clean. How is it? It's in bad shape. Weak from loss of blood.  Be quiet. But you've been in here five minutes and... he's just like he was before. Alex, you're stalling. But he's dying. I know. Alex, call the police. No. Are you going to let him die? Yes. That's as bad as murder. I didn't shoot him. It's not my fault if he dies. Lenore, he's got money. Thousands of dollars. No one knows he came here. This is our chancellor, Lenore. What do you mean?  Take his money, and then get rid of his body. There was a wallet containing a few one-dollar bills and the driver's license, which I handed to Lenore. His name is James Kirk. He lives in Hillsborough. His pockets were crammed full of bills of large denominations. Lenore watched, fascinated. How much is it, Alex? A little over $70,000. We're rich, Lenore. I'm frightened. There's nothing to be afraid of. All we have to do now is wait until he dies and get rid of the body. Do you need me to help you? No. What are you going to do now? It'd be dangerous to wait here until he dies. Anyway, what difference does it make? He'll be dead when he goes into the water. No. No. No, that's murder. All of what you like. We've gone too far now to stop. You wait in the car. I'm going to carry him to the edge of the pier and drop him over the side. Carry it out to sea. One doesn't picture a middle-aged drugist committing a perfect crime. But I felt sure that I'd done it. I had Kirk's money and it seemed that no one could connect his debt to me. I rejoined Lenore in the car and started for home. After I'd driven for a few minutes, I noticed she was strangely quiet. What's the matter then? Nothing. Nothing at all. Something's troubling you. What is it? Nothing. You act as though you're afraid of me. No. You are afraid of me. Lenore, why? I never saw you act this way before. Something's happened to you. You're different. Oh, nonsense. Don't you feel it, Alex?  Lenore. You wouldn't even wait until he was dead. Listen to me, Lenore. Whatever I did, I did for your sake. I wanted the money for you. So you could be happy. Oh, how can I be happy with this thing hanging over us? We had breakfast the next morning in absolute silence. Then Lenore went to open the store. Well, I wrapped the money in a parcel, took it to the bank, placed it in my box in the safe deposit ball. No one paid any attention to me. Although the bank was swarming with police detectives. From the bank, I went directly to the drug store and found a man there waiting. Sturges. Yes? Biggit is my name. Mark Biggit. What can I do for you? I'm an insurance company detective, Mr. Sturges. I'm investigating the first national bank robbery for my company. I want you to tell me everything you know about that robbery. But not bad. Although it didn't appear to be very intelligent. He had a look of cunning. His clothes were rumbled.  Suppose you start from the beginning, Sturges. There's nothing to talk about, Biggit. I don't know anything about the robbery. You don't, eh? No, I don't.  I'll tell you why I'm here, Sturges. And then maybe you'll see your way clear to talk. You know, the bank was robbed last night and the cook got away with better than $78,000. Yes? The night watchman was shot. He's in a coma now, but before he was shot, he shot one of the cooks. What does that have to do with me? I'm coming to you, Sturges. I'm a slow man, but I'm thorough. Which is more than you can say for the harness bulls and the police force. Think those harness bulls. They didn't spot the trail of blood that wounded crook left when he ducked on the alley behind the bank. Funny thing, they clean missed it. But I noticed it, Sturges. I followed it through the alley and straight up the street. And here's another peculiar thing. The trail ended right smack at the door of this drugstore. Are you implying... Two and two makes four, Sturges. As I said, I'm not a fancy pants detective, but here's how I figure it. This yag was wounded, so he came in here and asked you to fix him up, which you did.  Am I right, Sturges? You're 100% wrong. If you were smart, you'd talk now and save yourself a lot of grief. I've got nothing to tell you. Suit yourself, Sturges. But here's my business card. In case you change your mind, you can reach me. The Hotel Empire. By the way, the company is offering a $5,000 reward for the return of that money. I'm not interested. Good day. Okay. It's your next, Sturges. I'll be seeing. Alex, I heard what he said in those. I should have thought of the blood outside. How can you stand there and act so calm about it? He knows, I tell you. Take all of yourself in an awe. He only suspects. And his suspicions aren't worth two cents unless he can produce evidence. Believe me, we've got nothing to worry about. I was cocky then. Too cocky. And all was right. I was a changed man. That money had put iron into my spine. Then, that afternoon, I got the first of the phone calls. Hello? Sturges drug store? This is he speaking. Yes, of course. Alex, please. I'm frightened. Listen, Leno, calm yourself. It's perfectly clear what happened. One of the bank robbers came with Kirk as far as the store. He must have been frightened away by the police. Now he's using this method to get the money. But he won't get it. Why? What are you going to do? I'm going to put a water, blank paper and an envelope and mail it to that post office box. Then I'll stand watch at the post office and see who claims it. And then what? That money means everything to us, Leno. I've already killed one man for it. If necessary, I'll commit another murder. I placed blank strips of paper and an envelope and mail the envelope to that post office box. The next morning, I went to the post office and stood near the window, waiting for someone to come along and claim the envelope in box number 11. I was fast. I got hungry. My legs began to shake from exhaustion. I couldn't leave that spot. I went to stay there until... What are you waiting here for, Sturgis? The boys cut through my thoughts. I looked up. It was Kelly, the policeman. He was suspicious. Clerk noticed you hanging around and called for a cup. It's nothing, Kelly. I arranged to meet my wife here. She's late. I saw your wife through the drugstore window as I came up the street. You must have forgotten all about your appointment. You'd better get back to the store, Sturgis. I couldn't stay there after that. I walked around the block. When I came back, the policeman disappeared. I entered, went to the window, asked the clerk if anyone had claimed the mail in box 11. He nodded. The letter was gone. My plan to discover the identity of Kirk's confederate had failed. There was nothing to do but go back to the drugstore. I felt a sense of danger as soon as I entered. So I closed the street door quietly and walked as silently as I could to the door of the back room. I don't know. I was at the telephone. Hello. Hello, Empire. Hotel. Hello. Connect me with Mr. Bigot, please. Yes. Mr. Mark Bigot. Hang up. Hang up the receiver. Alex! You were calling that detective. Suppose I was. What of it? You were going to tell him about the money. You were going to double-cross me for the reward. You're so smart, Alex. What are you going to do about it? No.  I don't want the money for myself. I took it for you. To give you the things that will make you happy. Big hearted, unselfish, Alex. What are you laughing at? I'll tell you why you took that money, Alex. You thought you could buy me with it. When you're wrong. I wouldn't stay with you if you had all the money in the world. Oh, no. Would you leave me after all that's happened? I'm packing up right now. No. You can't stop me. I love you, Leno. I did my best to hold you. Can't have you. No one else will have you. What do you mean? Alex, stay away from me.  Alex, no.  I've... ...empty, drained all emotion. I buried her body in the cellar of the drugstore. I was even a bit glad that it ended in this fashion. For now and all. It would be with me. Did I say it was ended? I was wrong. I found no peace. For the very next morning, bigot came to the drugstore. Good morning, Sturges. Nice morning, isn't it? What do you want? Just stop by to have a chat, Sturges. We're giving a lot of thought to the First National Bank robbery. You know, that night Watchman came out of his coma this morning. Why tell me about it? I thought you might be interested. Here's an angle, Sturges. The Watchman says the robbery was pulled by only one man. One single crook, and he was shot. Nice angle, isn't it? Is it? Yes, it gave me a new idea, Sturges. I'm no Sherlock Holmes, but I figure it this way. That crook came in here and asked you to fix him up. You've gone through this before. I'm going to take it a little further, Sturges. The crook asked you to fix him up, but you've got to peek at all that money, so you let him die, took the money, and disposed of the body. How's that for you? Very clever. Can you produce the body? No, the police haven't been able to locate it, but I'm not the scourge, Sturges. Those harness bulls aren't very smart. I'll find that body. And when you do? I'm an insurance company, Dick, Sturges. I want that $78,000. That body, I'll come around here and you'll cough it up. I've told you before, I know nothing about that robbery or the money. Now I've got work to do. Both of you run along. Okay, Sturges. You're along. Sturges drugstore. This is he speaking. Read it to me, please. Is there a signature? No. My head was really... Out of the confusion in my mind, one thought emerged. The night watchman had said Kirk had no Confederate in the bank robbery. However, the person sending me the telegram must be Kirk. The salt water of the bay must have revived him.  I found Kirk's wallet on the shelf while I know I had placed it the night all this began. From the driver's license, I copied his home address. I waited till afternoon night. I drove to Hillsborough, where I found Kirk's apartment in an old tenement building. The door was unlocked. I drew out my gun. How did you know? How did I know you were going to come here, Sturges? Sit down in that chair by the table and I'll tell you. You sent the telegram. Yes, I sent them. You got me to think that Kirk was still alive. That's right. And you followed me here. You trailed me. It's all clear now, isn't it, Sturges? You take two and two and add it up and it comes out four. Now, where's the money? Money? We're going to cut out the cute stuff, Sturges. I want that money, now where is it? It's in a medicine jar on a shelf in the back room of the drugstore. All right, let's go, Sturges. First, you'll give me the money, then I'll take you down to police headquarters. We drove to the drugstore. I led them into the back room and knew exactly what I was going to do. And one of the shelves was a bottle of hydrochloric acid. Bigger thought it was the bottle containing the money. I reached up, uncorked the bottle, and plunged the fraying hook into a... No! I'm blind!  If I can only see you, where are you, Sturges? Sturges! Sturges, where are you? I was huddled against the wall. He stood between me and the door, screaming in pain, firing his gun blindly. The first three shots missed, but the fourth bullet struck me in the stomach. The last thing I remember is bigot running out of that room, shouting for help. I lost consciousness soon after. When I came to, I found myself here in this hospital. You know the rest of the story, doctor. That's your confession, eh? That's my confession. Did you get all that down on paper, Mr. Bigger? Got every word of it, doctor. Bigger, you... how did you get here? I thought hydrochloric acid... You thought you blinded me permanently, but not enough of the stuff to get into my eyes to do any real damage. I... no! Dead, Bigger. I've seen a lot of funny cases in my time, doc, but this speaks them all. He's a man who commits murder to get a lot of stolen money for his wife.  And the money... The money is safe and sound, and the bank's safe deposit ball. Exit Sturgeon.  The confession. Tonight must replay half-presentation. We'll be hearing from you again in the near future. Creep? Until next time. This is Hutter Galloway closing the door to the mystery playhouse, and reminding you to... sleep tight. Good night. Armed Forces Radio Service.


Let A contained in R. So, the following statements are equivalent. Then, the following statements are equivalent. So, what are the statements? 1, A is compact, 2, A is closed and bounded and 3rd one says A has property. And what is that property? We say every open cover of A has a finite sub cover. So, recall what was our definition of compactness? That was in terms of sequences. We said A set is called compact if every sequence has got a sub sequence which is convergent in the set. The second property closed and bounded. We defined closeness of a set again in terms of sequences. We said A set is closed whenever a sequence A n of elements of A converges somewhere, then the limit must be inside the set A. That was closeness. Bounded. That is interval of intervals. There is some interval big enough which contains the set A. That is boundedness. So, closeness and boundedness are mixture of some property of sequences and of property of intervals. Now, saying A has high number of property, it is just purely in terms of open sets. But recall what was an open set? A set is open if every point is an interior point. That is same as saying every point given any point in the set, there is an open ball around it which is inside the set. That was open set. Again in terms of sets, open balls, every open cover has got a finite sub cover. So, the important thing of this theorem is that later when there is no notion of sequences in higher courses in mathematics, there are courses where sequences are not not possible. That is called, if you happen to study what is called a course in topological spaces. There one defines only what are open sets. One does not go to sequences or anything. Defining is dealt in terms of declaring some sets to be open and then defining a set to be closed if its complement is open and then doing whatever we are doing. So, there the notion of compactness is defined in terms of Heine-Borel property. Because saying that a set is closed or bounded, closeness you can define in terms of openness, but boundedness requires notion of distance, boundedness. So, space is where this is not possible to define notion of distance, but we want to have a notion of compactness. So, there the definition is the third one that every open cover of that set has got a finite. So, that is taken as a definition in topological spaces of compactness. So, I am just giving a snapshot of something which you may come across. So, let us write a proof of this. One is equivalent to two that we have already proved that A is compact if and only if it is closed and bounded already proved. Let us prove two if and only if three. So, that we want to prove. So, let us look at suppose A has, I will just sort an Heine-Borel property as HB property. Given any covering has got a finite sub cover. We want to show A is closed and bounded. We want to show it as it is closed and bounded. So, let us first prove A is closed. It is bounded either way. It is easier to show bounded. So, let me just show that first. It does not matter actually because both have to be shown. So, A is bounded. Now, to show that A is bounded, A is a subset of, everything is in real line. By the way, A is a subset of real line. We are in the real line. A is bounded. That means it lies between bounds. Now, given any set A and I have to use somehow my tool that every covering has got a finite sub cover. So, keep these two things always in mind what you want to show and what is given. So, to show a bounded, I have to somehow manufacture some covering of the set A and use a fact that it has got a finite sub cover and conclude as a consequence that it is bounded. Now, to show it is bounded, if you can show A is between some minus n to plus n. Every A is a subset of the interval minus n to plus n that will be bounded. Now, once I want to show it is between minus 1 to n to n, can I sort of take that interval and make it a covering of the set A? How do I get a covering of, so that one of these elements is minus n to plus n? First of all, to apply Ionimbral property, I should have open interval. So, let us take the open interval minus n to plus n because if A is contained in minus n to plus n, then A is bounded obviously. How can you manufacture a covering so that minus n to plus n is one of the elements in that covering? I can let very n minus 1 to plus 1 minus 2 to plus 2 and go on expanding it. So, that will be a covering of the whole real line actually. In particular, it will also be a covering of the set A because A is subset of, so let me write, so note, so this is how one thinks of a proof. A is contained in union minus n to plus n, n equal to 1 to infinity which is contained in R, which is actually A is subset of R and this is actually equal to Y subset. This is equal to R. So, what you have done is we have given a covering of R and hence it is a covering of A also. But now this being a covering of, this is a covering of A and all elements are open intervals, so open sets. So, it is an open covering of A. So, by high-neighboral property, it must have a finite sub cover. So, let us apply that.  How do we think of the proof? We want to show A is bounded, lies between some limits. So, and that should be something like an interval and we should go to a covering. So, we manufacture a covering, so that this is a particular element and use any particular property. So, A is bounded. Let us show it is also closed. So, A is closed. Now, here lies a very nice idea as far as the notion of distance is concerned. See, we want to show A is closed. I can just give the proof directly and be happy about it. But I just want to give you a feeling for how does one think of a proof. If A is closed, how do I show something is closed? There are two ways. One, take a sequence and say it is converging somewhere and try to show that limit is inside. But how is that the convergence of a sequence will relate it to open coverings? My tool available is every open cover has got a finite sub cover. Somehow I have to bring in those things. To show closeness, I should bring in coverings somehow. Now, instead of showing A is closed, it is equivalent to showing A complement is open. So, here it is the idea. Why A complement is open is useful because openness is in terms of neighborhoods. Every point in A complement, if I want to show is open, then I have to show given any point in A complement, there is open interval around it. Somehow I am coming to some kind of a covering kind of a thing. So, let us use that definition. To show A is closed, I will try to show A complement is open. So, for that, let us take a point x belonging to A complement. To show, there exists some epsilon bigger than 0 such that x minus epsilon to x plus epsilon is contained inside A. So, that is what is to be shown. Sorry, A complement Now, how do I manufacture coverings from here? Every point I want to show this. Now, here is something which is very useful. So, here is a node. How to get open sets? So, look at x is a point. Look at all y belonging to real line. Say that the distance between x and y, x is a point. If x is less than or equal to, then it is a closed set. It is a closed ball. If I make it bigger than epsilon, what kind of a set is this? In the real line, x is fixed. I am looking at all y such that the distance of x and y is bigger. Obviously, it is a complement of the closed ball. So, it is an open set. You can look at that way. So, this set is equal to r minus y belonging to r. Say that x minus y is less than or equal to epsilon. That is a closed set. So, that means, so implies that this y belonging to r says that x minus y bigger than epsilon is a general fact. It is an open subset. You can do it also in Rn. In Rn also, it is a look at all the distance between x and y bigger than epsilon in Rn. That also is an open subset of Rn because only the notion of distance is required. Now, I have already got an open set. Now, I want to manufacture a covering. So, how do I go to a covering? This is happening for every epsilon. This is happening for every epsilon. So, how do I go to a covering? That is the question. For every epsilon, I want a family. How can I generate a family by varying epsilon? 1 over n. In particular, for every n, y belonging to r says that x minus y bigger than 1 over n is an open set. x is fixed. Is it okay? This is an open set. Also, what is the relation of this open set with a? Where was the point x? x was in a complement. So, what is the relation of this with a complement? Here is a complement. Can you say a complement? Here are the sets y belonging to r such that mod of x y minus x is bigger than 1 over n. So, somehow, what is my aim? I want to get a covering of, see, what is given to me? Given to me is, a is, I want to show a is close and given to me is, a is, a has a high borrel property. So, that means every, a has high borrel property. So, every covering of a has got a finite sub cover. So, what is the relation between, so, let me also, let us look at this family. These are sets where x belongs to a complement and n belongs to n. So, am I getting some kind of a covering of a from these things or not? Or a of a complement something? Are you able to see something or not? I have got a set a. I am looking at a complement. So, if point is, so, if x is in a complement, then all things which are bigger, they are open sets and x is in a complement. So, from here, I want a covering, so we want a covering of a from these things. So, what could be a family which will give me a cover of a? Getting something, some idea anybody has? See the, if I can show that the ball, see this is what is this? This is a ball, this is open set, open interval around x of radius of distance 1 over n. So, now, if I am able to show, so, if we are able to show that x belongs, see x is in a complement that is given to me. If x belongs to x minus 1 over n, x plus 1 over n, if x belongs to this, which is inside a, then I am trying to show that this kind of thing comes inside. So, I am trying to work backward now to indicate how is the proof going on, go on. Now, look at this x minus 1 over n to x plus 1 over n, this interval. If m is bigger than n, what is happening? If what is the relation between this for m bigger than or equal to n? What is the relation between these two intervals? Which one is contained? Because if m is bigger than n, then 1 over m is less than or equal to 1 over n. Both are centered at the point x. So, this is smaller than this one. So, let us say x minus 1 over m to x plus 1 over m is inside x minus 1 over n to x plus 1 over n, for every n bigger than or equal to n. That means the union of, if I take this union, m equal to 1 to n, x minus 1 over m, x plus 1 over m, actually that is equal to, I am just playing with intervals. What I want to show? I want to show this kind of a thing. I am just working out an argument which will work. So, I want to show this is contained in A. So, to show, this is what I want to show. If I revert it, that means what? A complement, I should be inside. So, where should be A complement? If these unions are inside A, where should be a complement? In the union, this is inside this. So, x minus 1 over m, x plus 1 over m complement. If something is inside A, complement is inside the complement of that. So, inside intersection of m equal to 1 to n, x minus 1 over m to x plus 1 over n, complement of that set. So, let me write complement of this interval. And what is this thing? So, this is equal to intersection m equal to 1 to n of, what is this thing? The same as y, such that x minus y is bigger than or equal to 1 over, 1 over, this is 1 over m. So, you do those, the sets coming again here. So, let us now go back. I want finite number of them should contain it. A complement should be inside finite number of them. So, let us go back. So, these were all open sets and here is, here is that. So, why this is an open set? And x belongs to it, singleton x belongs to it. So, A should be a finite union of, so we want to show it is open. So, every point is a neighborhood. If I put it bigger than or equal to, then it will be a closed set. So, for every x belonging to A complement, we are looking at the sets y belonging to R, such that y is bigger than or equal to, now mod of y minus x is bigger than or equal to, y is bigger than or equal to n, n bigger than or equal to 1. So, if x belongs to A complement and if I call these sets, so this is the ball centered at x of radius 1 over n and the closed thing. So, implies x belongs to this ball. I think I will have to postpone the proof because, see the basic idea is I want to show A is closed. That means it should be, can I use that it is open? It is open. I have to go to a covering of A. So, how to go to a covering of A? A minus, why do I vary? x belongs to A complement. I think x belongs to A complement that implies R minus a singleton x. A is a subset of, is it okay? If x belongs to A complement, A must be a subset of the complement of the singleton. Is it okay? Because x does not belong to A. So, even if I remove it, it is still a subset and what is this? This is equal to union. I think this seems to be okay. y minus x is bigger than or, why bigger than or equal to? Let us just write bigger than. I think this is what I was trying to reach 1 over n, n equal to 1 to infinity. Let us look at this statement. x belongs to A complement.  If I take any point in R minus x, then y is not equal to x. If I take any point in y in this, then that point is not equal to x. So, distance between x and y, absolute value of x and y should be bigger than something. So, for sufficiently large n, it will be bigger than 1 over n. So, I am saying these two are equal. Is that okay? This is equality. Is this equality okay for everybody? Yes. Because right hand side is a union of subsets of R. So, right hand side is a sub and x does not belong to it. So, that is inside this. Conversely, if I take a point y in R minus x, then this distance is something and that distance will be bigger than or equal to or bigger than 1 over n for some n. So, the left hand side is a subset of right hand side, right hand side is a subset of left hand side. Is that okay? Yes. So, a, now that is what I wanted to say that for a, I have got a covering now. A is compact and I have got, these are all open sets. We have already shown these are open sets. They are the complements of the closed intervals. So, there must be a finite sub cover. There must be a finite sub cover. So, implies by LUB property. I was just trying to motivate that this should be.  But R minus x now it is gone. We are directly using a is a subset of this union. So, that implies a is n equal to 1 to m, y mod y minus x bigger than 1 over m. Is that okay? Because not LUB, that Heine-Berl property. Sorry, what is given is Heine-Berl property.  We have complements of closed intervals. So, a must be covered by finitely many of them by Heine-Berl property. Compactness of, so if you see what I this is what I was trying to motivate you that a should be inside finite number a should be finite number of them anyway. So, this is I think sometimes trying to explain too much complicates the things. So, it is simplest is this is very simple. So, this implies what? Now, this bigger than 1 over m, bigger than 1 over n, n equal to 1 to m. But what is this right hand side? So, that implies a is a subset of what is this union? We just now observed for m bigger than n, 1 over this is a smaller one. So, what is the largest one? So, this is all y such that y minus x is bigger than 1 over n. 1 over n equal to 1 to m. So, this is m is 1 over m is the largest length possible and that implies now go to complement. What is the complement of this? That means, so y minus x less than or equal to 1 over m that set all y such that is a subset of a complement that implies the interval x minus 1 over m to x plus 1 over m, x plus 1 over m is inside a complement and x belongs to it. So, the crucial thing in this is only to observe that these are all mod y minus x bigger than epsilon for x fixed, epsilon fixed is an open set for every. So, that proves that a is also, so implies a is sorry not in a complement less than or equal to, so implies a complement is open. So, that proves. So, what we have proved is that if a has ain Borel property then it is closed and bounded. So, 3 implies 2, hence 3 implies 2. So, let us look at converse. What I want to prove? 2 implies 3. So, that means given is always good idea to write what is given. What is 2? They are given a is closed and bounded, a is closed and bounded to show a has ain Borel property. So, where do we start? To show it as ain Borel property I should start with an open covering of a and get a finite sub cover. So, let say u alpha, alpha belonging to i, b and open cover. So, I have to construct a finite sub cover. We only know a is closed and bounded. If a were a interval that we have already proved. If a were an interval we just now proved in the beginning that if a is a closed bounded interval then it has ain Borel property. But a may not be a closed bounded interval, but it is a closed bounded set. So, that means what? If it is a bounded set at least it will be inside a closed bounded interval. So, a bounded implies there exists n along the natural number such that a is inside minus n 2 plus n. Now, the aim should be from a covering of a, I should try to go to a covering of minus n to n and then use the theorem that for minus n to n I want a finite sub cover. So, from a covering of a I want to go to a covering of minus n to n. For a I already have a covering. For a I already have a covering. So, note a is inside union u alpha, alpha belonging to i. So, what I want? a is inside this, a is already covered and I also know a is. So, what is outside a? That is not covered. What is not outside? That is a complement. What can you say about a complement? What can you say about a complement? It is open. So, what is not covered by a? It is covered by a complement which is open. So, covering of a and a complement will give me a covering of everything in particular of minus n to plus n. So, that is the idea. A complement is open implies u alpha, alpha belonging to i. These sets, union with a complement is an open cover of, is open cover of r and hence of minus n to n, because minus n to n is a subset. Now, we use that fact, but close bounded intervals have high number of property since minus n to n has h b property. This is implies, I have got a covering. That means there is a finite sub cover. That finite sub cover of minus n to n may include a complement, may not include a complement, but positively it will have some finite sub collection of u alpha.  So, there exists u alpha m may be a complement cover minus n to n. From this collection, how do you get a finite sub cover? Keep a complement and pick up probably some finite number of elements from the given cover. So, that is what we have written. Now, if I delete from this, if I delete a complement, what will be that cover? This covers minus n to n, a is a subset of it. So, that implies what? The remaining one should cover a, because this is only covering a complement.  So, that proves that given a covering of a, we have got a finite. So, the idea is, boundedness puts it inside a interval and comp, collusion as gives me covering of complement. Put together, I get a covering. Select a finite sub cover of that interval minus n to n, remove a complement from that. Because we do not need, we need a covering of only a. So, remaining thing should cover a. So, that proves. So, hence we have proved that closed bounded implies high in Borel property. So, we have proved an important theorem namely for a subset of the real line. In this proof, 1 implies 2 and 2 implies 1. We did not use the fact that we are real, in the real line. That is anywhere it is to, R n is also true. Only proving 2 implies 3 and 3 implies 2, we heavily use the notion of distance that we are on the real line and the distance is given by the absolute value. So, this proof actually works for R n also. Because in R n, you can define notion of a sequence, closeness, openness, covering, open covering. But the proofs become slightly more involved. So, we will not be proving those theorems for R n. But with a remark that these theorems also, this theorem also remains true for R n. In fact, we will define later on what is called a metric space. Just we will define and is a generalization of R n, basically notion of distance and this theorem remains true even for metric spaces. But the proofs are slightly involved because there is no order, there is no way of saying bigger than or less than.