The two definitions of a compact set

Let $X \subset R$

1) Compact => bounded.

I find it easy to just do this. For every $x \in X$ let $V_x = (x-1/2, x + 1/2)$. $V_x$ is open and $X \subset of \cup V_x$. So {$V_x$} is an open cover. So it has a finite subcover. So there is a lowest interval and there is a greatest interval in the finite subcollection of intervals and X is bounded between them.

2) Compact => closed

Let X not be closed. Then there is a limit point,y, of X that is not in X. Let's let $V_n$ = {$x \in \mathbb R| |x - y| > 1/n$}. As this covers all $\mathbb R$ except $y$ and $y \not \in X$ it covers X. Take any finite subcover the is a maximum value of $n$ so $(y - 1/n, y + 1/n)$ is not covered by the finite subcover. As $y$ was a limit point, $(y - 1/n, y + 1/n)$ contains points of X. So the subcover doesn't cover X. So X is not compact.

Unfortunately Closed and Bounded => compact is much harder.

But I hope I gave you a sense of the flavor of compact sets.