Studying on my own currently and having trouble understanding a topology proof.

Theorem: Compact subsets of metric spaces are closed

Proof

Let be a compact subset of a metric space . We shall prove that the complement of is an open subset of

Suppose , . If , let and be neighbourhoods of and respectively, of radius less than

Since is compact, there are finitely many points in such that

If , then is a neighbourhood of which does not intersect . Hence , so that is an interior point of . The theorem follows.

Question

I am confused as to how and are defined

Is it that

Also, how does the chosen radius play a role in the proof?