1. Sequence and accumulation point

Suppose $\{a_n\}_{n\in\mathbb{N}}\to A$ and $\{a_n: \ n\in\mathbb{N}\}$ is an infinite set. Show that A is an accumulation point of $\{a_n: \ n\in\mathbb{N}\}$.

Let $A=\text{sup} \ \{a_n\}$ and Q be a neighborhood of A. There is an $\epsilon>0$ such that $(A-\epsilon, A+\epsilon)\subset Q$.
Since A = sup and if c is an upper bound, $A\leq c$
Let $a_i\in\{a_n\}, \ i=1,2,....$
Assume there are finitely many $a_i\in Q$.
Now, let $A'=\text{max}\{a_i\}$
Since x' is the max, x' is an upper bound of $\{a_n\}$. Therefore, $A', but $A = \text{sup} \ \{a_n\}$ which is a contradiction. Thus, A is an accumulation point.

Correct?

2. Re: Sequence and accumulation point

Originally Posted by dwsmith
Suppose $\{a_n\}_{n\in\mathbb{N}}\to A$ and $\{a_n: \ n\in\mathbb{N}\}$ is an infinite set. Show that A is an accumulation point of $\{a_n: \ n\in\mathbb{N}\}$.

Let $A=\text{sup} \ \{a_n\}$ and Q be a neighborhood of A. There is an $\epsilon>0$ such that $(A-\epsilon, A+\epsilon)\subset Q$.
Since A = sup and if c is an upper bound, $A\leq c$
Let $a_i\in\{a_n\}, \ i=1,2,....$
Assume there are finitely many $a_i\in Q$.
Now, let $A'=\text{max}\{a_i\}$
Since x' is the max, x' is an upper bound of $\{a_n\}$. Therefore, $A', but $A = \text{sup} \ \{a_n\}$ which is a contradiction. Thus, A is an accumulation point.

Correct?
No, that's not correct. You don't get to just assume that $A=\sup\{a_n\}$. For example $\langle (-1)^n/n\rangle$ is a sequence converging to $0$, which is neither an upper nor lower bound of the sequence.

Recall that by definition $\langle a_n\rangle\to A$ iff for each $\epsilon>0$ there is $N\in\mathbb{N}$ such that if $n\geq N$ then $a_n\in(A\pm\epsilon)$. So just notice that there are infinitely many elements in $\{a_n:n\geq N\}\subseteq(A\pm\epsilon)$, and the proof is complete.