We know that if $A$ and $B$ are compact (assuming A and B are non-empty), then the Cartesian product $A \text{x} B$ is compact. But how do you go the other way round.
We have to show that any sequence $(a_k)$ in A and $(b_k)$ in B have subsequences that converges in A and B respectively. We are given that any subseqence of the sequence $(a_k, b_k)$ is convergent in $A \text{ x } B$. I have at loss at how to I use this information to claim that subsequences of $(a_k) \text{and} (b_k)$ are convergent in $A$ and $B$ respectively.
Should I claim that $(a_k, b_k)$ is convergent iff each $a_k$ and $b_k$ is convergent, and be done with it?