Suppose $\Omega$ is a closed convex set in $\mathbb{R}^n$. Let $P_\Omega(u)$ represent the projection of $u$ onto $\Omega, \forall u \in \mathbb{R}^n$. That is to say $P_\Omega(u)= \underset{v \in\Omega}{\operatorname{argmin}}||v-u||$ for some norm $||\cdot||$. For your convenience, you may think $||\cdot|| $ as Euclidean norm.
How to prove $||P_\Omega(v)-P_\Omega(u)|| \le ||v-u||, \forall u,v \in \mathbb{R}^n$?
I have finished this proof by myself. The proof is a little bit tricky and requires some computation. I wrote it on my iPad so I hope it does not affect the readability.
I am not a native English speaker so if there are some grammar mistakes or wrong terms, you are welcome to correct them. BTW, you are also welcome to replace the pictures with Mathjax and Markdown.