The more simplistic idea is as follows...
Let be a topological space and a subspace. Then, is a compact subspace of if and only if it's a compact subspace of
To see this first assume that is a compact subspace of and let be an open cover with the subspace topology inherited from . Then, where is open in . But, note that and so has a finite subcover (since it's an open cover of as a subspace of ) from where the conclusion follows.
The other direction is similar.