Showing that if $P\in\mathcal{B}(\mathcal{H}):||P||\leq 1$ and $P$ is a projection mapping, then $P$ is necessarily an orthogonal projection

Question

Let $P \in \mathcal{B}(\mathcal{H})$ be a bounded projection mapping in an Hilbert space $H$. Suppose that $||P|| \leq 1$, i.e. $\forall x\in \mathcal{H}:||Px|| \leq ||x||$. I am trying to show that then $P$ is necessarily an orthogonal projection, but I am a bit confused on where to even begin. I know that $P$ is an orthogonal projection if and only if it is self-adjoint, for it is a priori a projection. Thus $\exists u\in \mathcal{H}:P(u)\neq P^*(u)$ so that

$$0 < ||(P^* - P)(u)||^2 = ||P^*(u)||^2 + ||P(u)||^2 - <P(u), P^*(u)> - <P^*(u), P(u)> \leq ||P^*(u)||^2 + ||u||^2 - <P(u), P^*(u)> - <P^*(u), P(u)>$$

But I don't really see how I can continue on from this.