Intuition behind compact subspaces of a metric space

Question

I've read up on compactness in a metric space and have found a few definitions (let $X$ be a metric space and $E \subset X$ in all the following):

1. $E$ is compact in $X$ if for every open covering of $E$ in $X$, $\{G_{\alpha \in A}\}$, $\exists$ a finite sub cover, $\{G_{\alpha \in B}\}$ s.t. $E \subset \cup_{\alpha \in B} G_{\alpha}$
2. $E$ is compact if every sequence in $E$ has a convergent subsequence.
3. $E$ is compact in $X$ if it is closed and bounded in $X$.

Now, I like the last definition better because it's easiest to understand, but I have a sneaking feeling they all are equivalent statements. From the first definition I can gather that the third definition follows, but I can't seem to wrap my mind on the idea that if $E$ is bounded and closed in $X$ then it has a finite subcovering in $X$. Then the second definition I can't really understand at all... I can't seem to intuitively relate it to either of the other definitions. I suppose my confusion stems from which of these definitions is best for describing intuitively what a compact subspace is?

EDIT: Changed the 2nd definition as the first comment points out that the original one was wrong

Answer 1

Your first definition is the standard one (both for metric spaces and topological spaces in general). Your third definition is true in Euclidean space (it's the Heine-Borel theorem), but not every metric space. An arbitrary metric space is sometimes defined to be compact if it is totally bounded and complete. Your second definition can be proved from this one. Note that definitions one and two are only equivalent in metric spaces (not in general topological spaces).