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Let () slence of real numheT Show that the subsequence (02 ) ((2,(14,06,u8. AND the subsequence Az Atli-(1.(5,(17' both convergetothe same number then the sque...

Question

Let () slence of real numheT Show that the subsequence (02 ) ((2,(14,06,u8. AND the subsequence Az Atli-(1.(5,(17' both convergetothe same number then the squence ( must convetge(Root Test) Assume that the following limit exists:lim Wonl

Let () slence of real numheT Show that the subsequence (02 ) ((2,(14,06,u8. AND the subsequence Az Atli-(1.(5,(17' both convergetothe same number then the squence ( must convetge (Root Test) Assume that the following limit exists: lim Wonl



Answers

Limits and subsequences If the terms of one sequence appear in another sequence in their given order, we call the first sequence a subsequence of the second. Prove that if two subsequences of a sequence $\left\{a_{n}\right\}$ have different limits $L_{1} \neq L_{2}$ then $\left\{a_{n}\right\}$ diverges.

Well, let's E M and I am B two but uh preserving deserving functions whose domains are the sets of positive in stages angles and who's ranges who's ranges uh, subsets I subsets of the positive, positive being teachers. So let's consider, So let's consider the two sub sequences that is E. N. And E I am where we have mm hmm mm. It's L one M E I am. It's L two But L one is not equal to go to here. So, so this implies that the absolute value of E. K. And my name's A I am Is the absolute value of L 1 -62 greater than zero. So, so they if you look at it there does that does not. Is this and such. That's for all um and Grace are done and big in. It implies bags we have the absolute value of E M minus its E. N. To be less than epsilon. So by exercise, 116 it implies that the sequence the sequence A. N. It's not convergence. It's not it's not convergence enhanced diverges has vivacious

So first time I would try to show that the Americans as I am and I would try to show that I am is bounded by three and grave for we have I want Nico just got up. Three will be smaller than three already. So that's fine in the Russian. I will write in black now, so I one is smarter than I want. Echo Ju scored a three citizen smarter than traces only drew here and now we should have I Ah, and bless one. We're smarter than I am because I am less one Jenny coach discovered in the three times I am And this one will be eagle juice carrot off. Uh, and great, we'll have this One will be smarter than equal Jew square, The three Temps three And it will be, Ah, Echo 23 dozen. We used the proof for induction because we assume in the stamp of us above that assume that I am will be smaller than three. So the next time you want to show that I am does one will be smaller than serious. Well, so that's why we will have to hear. So that far that I am would be smarter than three. Now, the next step I want to show that No, uh uh I am is increasing. And when the end is increasing, we go have dinner to show this one we're gonna have Ah, and is positive. And because I any smaller than three then we should have that I am this one. We go to the square root under three times I am And where it isn't gonna be big Good Dan, The square root of the Because three is bigger than I am. So we have a country place on a country place the tree by eye. And that's why we have the bigger here and now we have inside will be My answers isn't becomes I am. So therefore we show that I am This one is bigger than I am And now. Bye, Theo. Term that because I m a increasing is monotone, is it? Morning? Don't and bound it both. So doesn't in glass That I am is convergent. And in order to show the limit here, Yes, I have my Aniko juice, some letter am and then we have the I am. Ah, Well, uh, we will have, uh they'll hear where you go, Joe Square in the three times. Quinn. The three times going to treat at the top isn't Nico, Joe. Every boat Ah, does square root on the tree. I was signed now and that isn't good Ego Judah Scrotum. The three times I am and you know you could you know I am here. So we have done The airy coach discovered that three times out and we go have it somewhere it go to Ah, I was listening blind on every square echo 23 a. M and doesn't imply snap and square ministry. And we could use our our our fact around the air Ministry coaches are and doesn't oblige that Erica just on Erica to three. This one. We don't take the other coaches row because isn't the irresponsible tive. So they found a limit off the I am tickled your tree

So here we want to prove the limit of X squared. His expert zero equals. So let's set up some of our relationships using the definition of limit we have. Zero is less than X minus zero catch. The value is less than delta and we have halves. The value of X squared minus zero is less than excellent. We want to find a relationship between Delta and absolute. So first we can perform the subtraction of zero, which is just a simplify. We have absolute value of X is less than delta and we have the absolute value of X squared Less than excellent. No, we know that X squared is always positive because if x is a negative number, it becomes squared, it becomes positive. So the absolute value of X squared is simply X squared. Now we need to obtain the absolute value. Somehow we need to draw relationship so we can do over on the side is there is an identity square root of X squared is the absolute value of X because if you put a negative number index squared, it will become positive and then you square it and it goes back to X but it will be. It'll be positive even if you put a negative number in so we can use that and square both sides. And this gets us. The absolute value of X is less than the square root of absolute. Now we have some relationships that we can use because the absolute value of X is less in delta and the absolute value is less than the square root of excellent, so we can conclude that Delta equals the square root of absolute


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