Showing that every sequence has a convergent subsequence is probably the easiest way to do it. You would need to use the fact that a sequence in converges if and only if converges in and converges in .

On the other hand, it might be possible to use the definition directly: take an open cover of and try to show that it has a finite subcover. You can relate this to the compactness of and by using the projection maps: given by and given by .