Problem. Let $X$ be a Hilbert space and $\emptyset \neq K \subseteq X$ be closed and convex. Then, $$ \|P_Kx - P_Ky \| \leq \|x-y \|$$ for all $x,y \in X$. Here, $P_K$ is the projection from $X$ onto $K$; that is the unique nearest element in $K$.
It doesn't look too difficult but I can't really see how to prove it. Geometrically (in low-dimensional cases at least) it makes sense but I'm not sure how to connect these objects. How should I approach this?