How do I prove that the product of two closed sets is closed?

Question

Given $A \in \mathbb{R^q}$ and $B \in \mathbb{R^p}$ two closed sets, then $A \times B$ is a closed set in $\mathbb{R^m} = \mathbb{R^q} \times \mathbb{R^p}$

How do I prove the previous result by using sequences?

Answer 1

If you have a convergent sequence $(a_1, b_1), (a_2, b_2), \ldots$ in $A \times B$, then show the point of convergence is in $A \times B$ by looking at each component separately.