Show $\langle x-Px,y-Px\rangle \leq 0$ for $Px$ the projection of $x$ onto $C$ convex, and any $y \in C$.

Question

So far I did the following:

Write $x=w+Px$ for $w$ st. $Pw=0$. Then $\langle x-Px,y-Px\rangle=\langle w,y-Px \rangle$. From here I can intuitively see that is non-positive since 'the arrow' from $Px$ to $y$ is pointed inside $C$ and $w$ is at a right angle to the 'surface' of $C$ at $Px$ but obviously this is not a proof.

Can anybody give me a hint?