When is the supremum\infimum an accumulation point? Trying to show that if a sequence converges, it either has a maximum, a minimum or both, I reached a dead-end. Assuming it is not constant, it is still bounded and its supremum and infimum aren't equal. Then I assumed that the supremum and infimum are not in the sequence. 
I want to show that there are two subsequences that converge to each of them but for that to happen I have to show they are accumulation points. I tried to use definition but failed. I know logically that following my assumption they have to be accumulation points but I can't derive it from the definitions. Any help?

Consider the sequence $\{\mathbf{x}_n\}_{n\in\mathbb{N}} \subseteq A$, where $A \subseteq \mathbb{R}^n$. If this sequence is convergent, then it is bounded. Also, the sequence must converge to $\mathbf{x} \in \overline{A}$. Show that for all $\epsilon > 0$, there are finitely many $\mathbf{x}_n \notin N(\mathbf{x}, \epsilon) \cap A$ where $N(\mathbf{x},\epsilon)$ is some neighborhood of $\mathbf{x}$. Then, use the fact that a finite set is bounded and that $N(\mathbf{x}, \epsilon_0) \cap A$ is bounded for some $\epsilon_0$ to prove your claim.
