Prove that sequence of projections to nested subspaces is fundamental (= Cauchy sequence)

Question

Let $H - $ Hilbert space and $Y_0 \supset Y_1 \supset Y_2 \supset \cdots - $ subspaces in $H$, $x_0 \in H$. Define a sequence $$x_{n+1} := P_{Y_n} x_n = P_{Y_n} P_{Y_{n-1}} \ldots P_{Y_0} x_0 = P_{Y_n} x_0.$$ Prove that this sequence $\{x_n\}$ is fundamental (=Cauchy) sequence, i.e $|x_n - x_m| \rightarrow 0$ when $n,m \rightarrow \infty$.

Answer 1

Put $$z_{n} = x_n - x_{n+1} = x_{n} - P_{Y_n} x_n$$ Hence $z_{n} $ and $Y_n $ are orthogonal. Moreover, $z_n = P_{Y_{n-1}} x_{n-1} - P_{Y_n} x_n$ and thus $z_n \in Y_{n-1}$.

We have

$$x_n = (x_{n} - x_{n-1}) + (x_{n-1} - x_{n-2}) + \ldots + (x_{1} - x_0) + x_0 $$ $$= -z_{n-1} - z_{n-2} -\ldots - z_1 + x_0.$$

It follows that $z_{n-1} + z_{n-2} + \ldots + z_1 = x_0- x_n = x_0-P_{Y_{n-1}} x_{0} $ and hence $z_{n-1} + z_{n-2} + \ldots + z_1$ and $Y_{n-1}$ are orthogonal.

We have $$x_0 = x_n + (z_{n-1} + z_{n-2} + \ldots + z_1 ),$$ where $z_{n-1} + z_{n-2} + \ldots + z_1$ and $Y_{n-1}$ are orthogonal and $x_n = P_{Y_{n-1}} x_{0} \in Y_{n-1}$. Thus $$||x_0||^2 = ||x_n||^2 + ||z_{n-1} + z_{n-2} + \ldots + z_1 ||^2.$$

Similarly we get that all $z_i$ are orthogonal (e.g. $z_1$ and $Y_1$ are orthogonal and $z_2, \ldots, z_{n-1} \in Y_1$ and so on) and $||z_{n-1} + z_{n-2} + \ldots + z_1 ||^2 = \sum_{i=1}^{n-1} ||z_i||^2$.

Hence $$||x_0||^2 = ||x_n||^2 + \sum_{i=1}^{n-1} ||z_i||^2$$ and it follows that $ \sum_{i=1}^{n-1} ||z_i||^2 \le ||x_0||^2 $ and thus $$ \sum_{i=1}^{\infty} ||z_i||^2 \le ||x_0||^2 < \infty .$$

For $m \ge n$ we get $$||x_n - x_m||^2 = || (z_{n-1} + z_{n-2} + \ldots + z_1 + x_0) - (z_{m-1} + z_{m-2} + \ldots + z_1 + x_0) ||^2$$ $$= ||z_{n+1} + z_{n+2} + \ldots + z_{m}||^2 = \sum_{i=n+1}^{m} ||z_i||^2 \to 0$$ as $n,m \to \infty$, q.e.d.