Let $\displaystyle H$ be a Hilbert Space and $\displaystyle P$ and $\displaystyle Q$ be the projections on closed linear subspaces $\displaystyle M$ and $\displaystyle N$ of $\displaystyle H$.

Prove that if $\displaystyle Q-P$ is a projection,then its range is $\displaystyle N\cap M^\perp$.

My attempt:

Let $\displaystyle x\in$ range of $\displaystyle Q-P.$

Since $\displaystyle Q-P$ is a projection,

$\displaystyle \Rightarrow(Q-P)x=x$

$\displaystyle \Rightarrow Qx-Px=x$

I am stuck here as i don't know how to proceed to show that $\displaystyle x\in N$ and $\displaystyle x\in M^\perp$

Can anyone give me some hints to proceed?

If I am in the wrong way to prove the question,please tell me.