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Projection onto a convex set

I have been tasked with the following problem:
Let $S\subset \mathbb{R}^n$ be a convex set and let $x\in\mathbb{R}^n$ be such that $y\leq x$ for all $y\in S$. Define $P_S(x)=\sup \{y\in S:y\leq x\}$. Show that $P_S$ is a projection from $\mathbb{R}^n$ onto $S$.
Attempt:
Let $\mathbf{v}\in \mathbb{R}^n$. If $v\geq P_S(x)$, then $v\geq \sup\{y\in S:y\leq x\}=x$. This implies $\langle \mathbf{v},\mathbf{v}-x\rangle\geq 0$. Therefore, $P_S(x)$ is the projection of $x$ onto $S$. Now, to prove that $P_S$ is a projection, we need to show that $\|P_S(x)-P_S(y)\|\leq \|x-y

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