theorem, Cartesian product of compact or convex sets

Theorem

If $A \subset \mathbb{R}^n$ and $B \subset \mathbb{R}^m$ are compact sets, then the set $A \times B$ is a compact subset of $\mathbb{R}^{n + m}$. If $A$ and $B$ are convex sets, then $A \times B$ is a convex subset of $\mathbb{R}^{n + m}$.

References

1. M. Maschler, E. Solan, and Shmuel Zamir, Game Theory, Cambridge University Press, 2013, p. 167.