Can anyone provide a proof that a compact set in metric space $(X,d)$ is bounded using..

Question

using anyone of the following definitions(and no other concerning compactness):

-$A \subseteq (X, \tau)$ is compact if for every open cover of A there exists a finite cover.

-A compact set in a Hausdorff space is closed.

-A closed subset of a compact set is compact.

-A compact set in metric space (X,d) is closed.

I tried to make something happen. Nothing. Search ME, nothing again.

Answer 1

Let $x\in A$. Cover $A$ with the open cover $\bigcup_{n\geq 1}B_n(x)$ where $B_n(x)$ is a ball of radius $n$ centered at $x$. Then what does it mean to have a finite subcover?