Sequence of projection is Cauchy

[yifli]

Homework Statement

Let {M_i} be an orthogonal sequence of complete subspaces of a pre-Hilbert space V, and let P_i be the orthogonal projection on M_i. Prove that {P_i(e)} is Cauchy for any e in V


2. The attempt at a solution
I'm trying to prove as n and m goes infinity, $$\left\|P_n(e)-P_m(e)\right\|^2\rightarrow 0$$

Here is what I've got so far:
$$\left\|P_n(e)-P_m(e)\right\|^2=\left\|P_n(e)\right\|^2+\left\|P_m(e)\right\|^2$$ because P_n(e) is orthogonal to P_m(e), how to proceed?

 

[micromass]

Hi yifli!

First, we can take V to be a Hilbert space. Indeed, try to prove this by taking the completion of V.

Second, try to do something with $\sum_{i=1}^{+\infty}{P_i(e)}$?

 

[yifli]

Second, try to do something with $\sum_{i=1}^{+\infty}{P_i(e)}$?

I did try that:
$$P_n(e)=e-\sum_{i=1,i \neq n}^{+\infty}{P_i(e)}$$.

substituting the above formula into $$\left\|P_n(e)-P_m(e)\right\|$$ gives me the same thing

What did I miss?

 

[micromass]

I did try that:
$$P_n(e)=e-\sum_{i=1,i \neq n}^{+\infty}{P_i(e)}$$.

Firstly, This formula doesn't hold.
Secondly, my point was that $\sum_{i=1}^{+\infty}{P_i(e)}$ converges. Thus...