in my Analysis course we saw this statement :
We are given $U,V$ two metric spaces and $k$ the distance
Let $U,V$ compact sets then $\; \longrightarrow \; $$ U \times V $ is also compact
Our teacher told us that we can prove it using compactness by the Bolzano-Weierstrass condition.
So here is what I know , A metric space is compact if it has the B-W Property.
I didn't found any definition in our lecture so I took the following :
Definition: A set S in a metric space has the Bolzano-Weierstrass Property if every sequence in S has a convergent subsequence — i.e., has a subsequence that converges to a point in S.
I don't know how I can apply it to my problem , so thanks in advance for your help.