Let $K$ be a nonempty compact convex subset of $\mathbb R^n$ and let $f$ be the function that maps $x \in \mathbb R^n$ to the unique closest point $y \in K$ with respect to the $\ell_2$ norm. I want to prove that $f$ is continuous, but I can't seem to figure out how.
My thoughts: Suppose $x_n \to x$ in $\mathbb R^n$. Let $y_n = f(x_n)$ and let $y = f(x)$. By the compactness of $K$, there is a convergent subsequence $(y_{k_n})$ that converges to some y' \in K. If y \ne y', then \|x-y\| < \|x-y'\|. Furthermore, any point $z \ne y$ on the line segment joining y,y' also satisfies $\|x-y\|<\|x-z\|$. I don't know where to go from here. Any tips?