Linked Questions

1
vote
1answer
127 views

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 ...
0
votes
1answer
30 views

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 ...
9
votes
5answers
732 views

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 ...
6
votes
3answers
7k views

A subsequence of a convergent sequence converges to the same limit. Questions on proof. (Abbott p 57 2.5.1)

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}+}...
-1
votes
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 ...
2
votes
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 ...
3
votes
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 ...
2
votes
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 ...
3
votes
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 ...
1
vote
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?
0
votes
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 ...
1
vote
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 $\mathbb N$, ...
0
votes
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 \...
0
votes
2answers
82 views

How to prove limit of sequences formally.

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 ...
1
vote
1answer
91 views

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 ...

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