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### If ${a_n}$ converges, then ${a_{2n}}$ converges proof question [duplicate]

I am trying to prove that if a sequence $\{a_n\}$ converges, then the sequence $\{a_{2n}\}$ converges as well using the definition of convergence. What I have so far is if $\{a_n\}$ converges, say ...
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### Find limit of a sequence $Y_n$ when $Y_n = X_{2n}$ and limit $X_n=l$ [duplicate]

I'm stumped by the following question. Can anyone help me out. Clearly the answer is l but I can't work out how to prove this Suppose that $\{x_n\}$ is a sequence converging to a limit $l$. Define a ...
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### Stuck at proving whether the sequence is convergent or not

I have been trying to determine whether the following sequence is convergent or not. This is what I got: Exercise 1: Find the $\min,\max,\sup,\inf, \liminf,\limsup$ and determine whether the ...
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Solutions to Homework 3 doesn`t duplicate. We have to prove that if $(a_{n})$ is a sequence in $\mathbb{R}$ with $\displaystyle \lim_{n\rightarrow\infty} a_n =a$, and if $(a_{n_{k}})_{k\in \mathbb{N}+}... 2answers 1k views ### If a sequence converges, then every subsequence converges to the same limit — but how do I know a subsequence exists? I have been reading the following post: Prove: If a sequence converges, then every subsequence converges to the same limit. I understand the idea, but I wonder, does this proof imply that such a ... 5answers 222 views ### Limit of$\sin(1/x)$- why there is no limit? $$\lim_{x\to 0+} \sin\left(\frac{1}{x}\right)$$ I know that there is no limit. but, why there is no limit? I tried$x=0.4$,$x=0.3$,$x=0.1$, it looks like the limit is$0$. And how can I show that ... 3answers 514 views ### Prove by counterexample that a bounded sequence in a metric space need not have a convergent subsequence I am trying to prove that a bounded sequence in a metric space need not have a convergent subsequence. My counterexample is: Consider the metric space: ($(0,1]$,standard norm on$\Bbb R$restrict to ... 3answers 747 views ### If the sequence$(a_n) $converges, then$(a_{n-1}) $also converges to the same limit as$(a_n) $If the sequence$(a_n)_{n \in \mathbb {N}} = (a_n)$converges to the real number$L $, then the sequence$(a_{n-1})_{n \in \mathbb {N}} = (a_{n-1}) $also converges to$L $. This appears true ... 2answers 365 views ### Proving in the formal way that a sequence is divergent I have the following sequence (I've already put it in the$a_n - L$form) $$\left|\frac{(-1)^n n+1}{n + 2}-L\right|\ge\epsilon$$ I think that in order to prove that the sequence is divergent I need to ... 5answers 123 views ### Is it true that$\overline{A}\cup\overline{B}$=$\overline{A\cup B}$?$\overline{A}$denotes the closure of the set$A$. I can show that$\overline{A}\cup\overline{B}\subset \overline{A\cup B}$, but I don't know how to go about the reverse direction? 0answers 551 views ### If a sequence converges, then every subsequence converges to the same limit Prove: If a sequence converges, then every subsequence converges to the same limit. Instead of saying that$n_{k}\geq k> N\implies |a_{n_{k}}-L|<\epsilon$Can i say that let$i$be the ... 2answers 59 views ###$(a_{(k_n)})_{n\in\mathbb N}$converges to L? The problem: Suppose$(a_n)_{n\in\mathbb N}$is a sequence in$\mathbb R$moreover that$(a_n)_{n\in\mathbb N}$converges to$L \in \mathbb R$.$(k_n)_{n\in\mathbb N}$is a sequence in$\mathbb N$, ... 3answers 111 views ### Find${\lim_{(x,y)\rightarrow \infty}}f(x,y) = \frac{2x + 3y}{x^2+xy+y^2}$. I have tried to solve$\lim_{x\rightarrow \infty ,y\rightarrow\infty}f(x,y)$, where$f(x,y) = \frac{2x + 3y}{x^2+xy+y^2}$. Can I define$y = r \sin \theta$and$x=r\cos \theta$when$x\rightarrow \...
How do I prove $\lim_{n\to\infty} a_n = \lim_{n\to\infty} a_{n+k} \text{ and } a_{n+1}$ is convergent, where $k$ is a fixed natural number and $a_n$ is a convergent sequence? I know I need to use the ...
### is the sequence $\Big(1+\frac{(-1)^n}{n}\Big)^{(-1)^nn}$ divergent?
is this sequence : $\Big(1+\Big(\frac{(-1)^n}{n}\Big)\Big)^{n(-1)^n}$ divergent ? I notice that when $n$ is even it converges to $e$ and when $n$ is odd is converges to $e$ as well but I've seen ...