Prove that a compact subset, $S$, of a normed vector space, $V$, is closed and bounded.
Attempt:
1) Showing $S$ is closed
To show $S$ is closed we have to show that $S$ contains all of its limit points.
PF: Suppose there exists a sequence $\{s_{n}\}_{n=1}^{\infty} \in S$ such that $s_n \rightarrow s$. Where $s$ is a limit point of the set $S$. I will show that $s \in S$.
Since $s \in V$ and we assumed $s_n$ converges this means $$\lim_{n \rightarrow \infty} \|s_n - s \| \rightarrow 0$$
Since $S$ is compact this means every sequence $s_n \in S$ has a subsequence $s_{n_{k}}$ that converges to a point $d \in S$. This also means $$\lim_{k \rightarrow \infty}\|s_{n_{k}} - d\| \rightarrow 0$$
Consider $\|s-d\|$:
$$\|s-d\| = \|s- s_{n_{k}} + s_{n_{k}} - d\| \leq \|s- s_{n_{k}}\| + \|s_{n_{k}} - d\| $$
Since $s_n \rightarrow s$ all subsequences also converge to $s$. This means $\lim_{n \rightarrow \infty}\|s- s_{n_{k}}\| \rightarrow 0$. As well We have also established that $\lim_{k \rightarrow \infty}\|s_{n_{k}} - d\| \rightarrow 0$.
Therefore: $$\lim_{k \rightarrow \infty}\|s-d\| = \lim_{k \rightarrow \infty}\|s- s_{n_{k}} + s_{n_{k}} - d\| \leq \lim_{k \rightarrow \infty}\|s- s_{n_{k}}\| + \lim_{k \rightarrow \infty}\|s_{n_{k}} - d\| = 0 $$
So $s = d \in S$. Therfore $S$ is closed.
2) Showing $S$ is bounded.
To show $S$ is bounded, I want to show that for all $s \in S$ there exists $M$ such that $|s| \leq M$.
PF: Towards contradiction suppose $S$ was unbounded. given that $S$ is compact, there exists a convergent subsequence $s_{n_{k}}$ that converges to a value $s$ in $S$. Therefore: $$|s| = \lim_{k \rightarrow \infty}\|s_{n_{k}}\| \geq \lim_{k \rightarrow \infty} = +\infty $$.
This is not possible thus a contradiction.
Comment: For showing closed I know that is the right procedure, but I feel I may have messed up in expressing the convergence of the two sequences using $k$. Feedback about the proof?