For metric spaces there is another, perhaps more natural way of thinking about compactness. It is based on the following classical result.
The Bolzano-Weierstrass theorem
Every bounded sequence in R has a convegent subsequence.
This is attributed to the Czech mathematician Bernhard Bolzano (1781 to 1848) and the German mathematician Karl Weierstrass (1815 to 1897).
From this we are led to the generalisation:
A metric space is sequentially compact if every bounded infinite set has a limit point.
The main result is:
A compact metric space is sequentially compact.
Let A be an infinite set in a compact metric space X. To prove that A has a limit point we must find a point p for whicch every open neighbourhood of p contains infinitely many points of A.
Suppose that no such point existed. Then every point of X has an open neighbourhood containing only finitely many points of A. These sets form an open cover of X and extracting a finite open cover gives a covering of X meeting A in only finitely many points. This is impossible since A X and A is infinite.
In a compact metric space every bounded sequence has a convergent sub-sequence.
Given the above limit point p, take xi1to be in a 1-neighbourhood of p, xi2to be in a 1/2-neighbourhood of p, ... and we get a sub-sequence converging to p.
Together with the Heine-Borel theorem this implies the Bolzano-Weierstrass theorem.
In fact, a metric space is compact if and only if it is sequentially compact. The proof that sequentially compact comact is harder.