Let $H$ be a Hilbert space with inner product $\langle\cdot,\cdot\rangle$ and let $V,W$ be two closed subspaces. For $x_0\in H$ we may define the sequence of projections $x_{2n+1}=P_W(x_{2n}), \qquad x_{2n+2}=P_V(x_{2n+1}).$ I intend to prove that if $V\cap W=\{0\}$, then $x_n\rightarrow 0$.
In the first place, one can easily show that $||x_{n+1}||^2=\langle x_{n+1},x_n\rangle$, from which one easily deduces that the sequence of norms is decreasing, so that $\lVert x_n\rVert\to\ell\geq 0$.
Then one can also show that if $V\cap W=\{0\}$ and a subsequemce $\{x_{2n_k}\}_k$ converges weakly to some $x$, then the sequence $\{x_{2n_k+1}\}_k$ also converges weakly to $x$, whence $x=0$.
So now it suffices to prove that a subsequence $\{x_{2n_k+1}\}$ converges weakly to something. For this, I first note that $\lVert x_n\rVert^2=\langle x_{n+1},x_{n-2}\rangle=\cdots=\langle x_{2n-1},x_0\rangle$ And now I would like to use that the sequence of norms converges and Riesz's representation theorem to conclude weak convergence but I am unsure as to how to proceed, since while it is true that any functional in $H$ is of the for $\langle \cdot,x_0\rangle$, varying $x_0$ also varies the sequence $\{x_{2n-1}\}$.
Thanks in advance for any insight.