Let $C$ be the Cantor set, $S:=C\times C$. Given: $C\times C$ is compact. Prove that $\exists x \in S$ such that $\lVert k - x \rVert \le \lVert k-y \rVert$, for $x,y \in S$ and $k$ a fixed point in $\mathbb{R}^2\backslash S$.
Proof:
Fix $k\in \mathbb{R}^2\backslash S$ and define the function $f_k:S\to \mathbb{R}$ as $f_k(x)=\lVert k-x\rVert$. Then $f$ is continuous, since norms on $\mathbb{R}^n$ are continuous. Since $S$ is compact, by the generalized Extreme value theorem, $m:=\inf(f_k) \le f_k \le M:=\sup(f_k)$. This implies that $\lVert k-x\rVert\le M=\lVert k-y \rVert$ for all $y\in S$. Hence, there is a point $x\in S$ which is closest to an arbitrary point $k\in \mathbb{R}^2\backslash S$.
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