I will take a remedial exam for Real Analysis and studying the concepts again. So I tried to proof Heine-Borel Theorem but there is some problems and I need help or verification. There are some proofs, yet they all are in general topological space which I can not understand with my current knowledge.
Heine-Borel Theorem in Tao Analysis II: Consider (${R^n},d$) where d is one of the metrics $d_{l^1},d_{l^2}, d_{l^\infty}$. A subset $E \subset\ {R^n}$ is compact if and only if E is bounded and closed.
Proof: $\Leftarrow$ Assume E is bounded and closed. Let $x_n$ be a sequence in E. Since E is bounded, $x_n$ converges thus has a convergent subsequence which converges in E, since E is closed (E contains limit points). Since we did not specify $x_n$, it is arbitrary and every sequence has convergent subsequence, thus E is sequentially compact. This implies compactness, thus E is compact.
$\Rightarrow$: E is compact, means that sequentially compact, implies every $x_n$ in E has a convergent subsequence. This is where I got total lost. I have ideas about it but can not construct the formal way. (I will add them to comments not to make the question too long.)
Also, we know that ${R^n}$ is closed under one of those metrics (We can assume that, I know how to prove it). I don't know if this is useful for this proof, but you can assume that I know that if necessary.