I am reading the proof of the theorem $22$ on page $13$ of this lecture notes. How Bessel's inequality was used to prove the convergence of the sequence $(P_mu)$ in $V$ where $V$ is a Hilbert space? I am asking this because there exists a sequence in a Hilbert space that converges weakly, but not converges strongly, then the Bessel's inequality does not seem sufficient to me to ensure that the sequence $(P_mu)$ converges in $V$.
Thanks in advance!