Show that the norm of this operator is equal to 1

Question

Let $H$ be a Hilbert space and $P$ a projection $H \rightarrow H$ ( a bounded linear operator on $H$ such that $P^2=P$ and $P$ is not equal to $0$)

I showed that $ ||P|| \ge 1$ and that $P$ is auto adjoint. I also know that $||P^*P|| = ||P||^2$ where $P^*$ is the adjoint of $P$

How can I now show that $||P||=1$ ?

Answer 1

I think I found it.

$||P^*P|| = ||P^2||= ||P||$

but also : $||P^*P||=||P||^2$

so, since $||P|| \ge 1$, it must be equal to $1$