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Claim: If $X \subset \mathbb{R}^n$ is compact, then $X \times X \subset \mathbb{R}^n \times \mathbb{R}^n$ is compact. Key definitions: A set $X$ is closed if every convergence sequence converges to a point in the set $X$. A set $X$ is bounded if there exists $M > 0$ such that for all $x \in X$,...