Let $\{P_n\}_n$ be a countably infinite collection of unique orthogonal projections in a Hilbert space $H$ and $\psi\in H\setminus \{0\}$. Then why does the sum $\sum_{n=0}^\infty P_n\psi$ converge in norm? It is certainly the case that $||\psi||^2 = \sum_{n=0}^\infty ||P_n\psi||^2 < \infty$ as per the Pythagorean Theorem, but I don't see why this would show that $\sum_{n=0}^\infty ||P_n\psi|| < \infty$.
I sort of understand the motivation for the claim: If $n_1,n_2,\dots$ are such indices that $P_{n_k}\psi\neq 0$, then $\psi = \sum_{k=1}^\infty P_{n_k}\psi$ and as each $P_{n_k}$ is an orthogonal projection we have that $\psi = P_{n_1}\psi + P_{n_1}^\perp\psi, P_{n_1}^\perp\psi = \sum_{k=2}^\infty P_{n_k}\psi$ etc. That is, the "mass" of $\psi$ is spread out on different orthogonal parts of the space $H$ and there is a finite total sum of the mass. As each different component is orthogonal to each other, some component $P_{n_k}\psi$ cannot affect $P_{n_l}\psi$ for $k\neq l$. But this does not constitute to a proper proof.