Orthogonal projection: more linear algebra questions

Orthogonal projection: more linear algebra questions

I'm working on proving the following:
If $P:V\to V$ is a linear operator and $P^2=P$ and $\|Pv\|\le\|v\|$ for
all $v \in V$ then $P$ is an orthogonal projection.
My thoughts on this are that a linear map with the following properties is
an orhtogonal projection: $null(P) $ is the orthogonal complement of its
range, $P^2 = P$ and $\|Pv\|\le\|v\|$.
Therefore it should be enough to show that the range of $P$ is orthogonal
to its null space. How to do this? Let $n \in null(P)$ and $u \in range(P)
= U$. Want to show: $\langle u,n \rangle = 0$.
Am I doing this right? And how can I show $\langle u,n \rangle = 0$?