Norm limit of sequence of orthogonal projections on Hilbert space "contractive"

Question

Let $(P_n)_{n \in \mathbb N} \subseteq B(\cal H)$ be a sequence of (orthogonal) projections on a (separable) Hilbert space such that $\left\Vert P_{n}\xi\right\Vert \rightarrow C\left\Vert \xi\right\Vert $ for every $\xi \in \cal H$, where $C \leq 1$. It feels like $C$ should be $1$ since $\left\Vert P_{n}\right\Vert =1$ for every $n$, but I don't see how to prove this. Is my guess even right?

Answer 1

If $\{e_n\}$ is an orthonormal basis for a Hilbert space and $P_n$ is the projection on the closed subspace spanned by $\{e_n,e_{n+1},\cdots \}$ then $\|P_nx\| \to 0$ for every $x$. So $C$ can be $0$.