This problem covers limits. In order to solve part A and B of this problem, we will use the approach of contradiction. Let's look at part A. First. The claim is that if S. N. Is greater than equal to A. For all but finite lee many N than limit, S N is greater than equal to A. Let us assume for a contradiction that this limit Yeah, is less than let us assume. And let's say absalon is equal to a minus S. This is the assumption. By the definition of limits, we can say that. Bye. Definition of limit. We can see that there exists capital N. Substantive small and is greater than capital in then sn minus S is less than epsilon. This is what the definition says. I'll write it down there exists capital and such that Yeah. If and is greater than capital N. Then and send minus S. Yes, less than absolute. So if you open this inequality, you know, you will get minus epsilon Nice between s and minus S taking minus s on both sides. You get s minus absalon as less absalon. Now, you can see here that S plus epsilon is nothing but A. So you will use that factor and I can say that s minus epsilon. Uh it means that essen belongs to s minus absalon. Call my A. What does this mean? It means that S. N. Is less than a, S. N is less than A. But our problems said that our assumption was based that S and will be greater than equal to way, but we got the opposite. It means that for all but finite lee many end S. N. Is less than in this problem, as per the assumption, which is a contradiction. It means that our initial claim was correct that if S. N. Is greater than equal to A. For all but financially many in limit S and will be greater than equal to It similarly will solve part B of the problem. Part B says that if S. N is less than equal to be for all, but finitely many end then limit essen is less than equal to be. Let us assume again for contradiction. We will assume, Oh, that limit essen is greater than B. We're assuming as a contradiction. And let us take absalon as S minus B. Again, by the definition of limit there exists and such that if N is greater than in the sm minus S is less than absolute, just like this. We use their definition of LTD so we get yes and minus S is less than epsilon. Again, opening the inequality minus epsilon, S n minus S rather than absolute, this becomes s minus epsilon. Essen? S less. Absolutely. From here, we can see that S many cepsa loans nothing but B. So I can see here, B essen less than s plus absalon. So from here it is clearly seen that sns greater than B. Or I can say for all. But finitely many N. We have S. And greater than B, which is a contradiction according to our statement. Because we took us and less than equal to be. Hence the claim that if S. N. Is greater than less than equal to be for all. But finitely many end then limit us and less than equal to be was correct. And contradiction proved it false. Let's go to part C. Part C is a direct conclusion from part A and part B from part A. You know that? Yes. Which is the limit limit of essen is greater than equal to A. And from part B. This limit S it's less than equal to me, so S belongs to it to be. That's all. Yeah.