How to show a bound on expected sum of squared martingale increments? I am reading a paper with the following set up. Let $(\Omega, \mathcal{F},P)$ be a probability space, $B \in \mathcal{F}$, and $\{\mathcal{F}_n : n \in \mathbb{N} \}$ a filtration with $\mathcal{F}_n \uparrow \mathcal{F}$.
Let $q_n = P(B \mid \mathcal{F}_{n-1})$. The sequence $\{q_n \}$ is a martingale with values in $[0,1]$ almost surely. Now, the paper asserts
$$E\Big(\sum_{n=1}^\infty(q_{n+1} - q_n)^2  \Big) \leq 1.$$
I can't see why this is true. Of course, for all $n$, $E(q_{n+1}-q_n)^2 \leq 1$, but I don't see why the inequality should hold after summing over all $n$. Could someone please point out what I'm missing?

Increments of a martingale are orthogonal hence 
$$\mathbb E\left[\sum_{n=1}^{+\infty}\left(q_{n+1}-q_n\right)^2       \right]
=\mathbb E\left[\left(\sum_{n=1}^{+\infty}q_{n+1}-q_n\right)^2    \right].$$ By the martingale convergence theorem, $$\sum_{n=1}^{+\infty}q_{n+1}-q_n=\mathbb P\left(B\mid\mathcal F\right)-\mathbb P\left(B\mid\mathcal{F} _0\right),$$
which gives the result.
