Theorem 12.2 (b) of Williams' "Probability with Martingales" states that the convergence of a series of zero-mean independent and bounded random variables $X_k$ (say bounded by $K$) implies the convergence of the series of their variances. That is, if $\sum_{k} X_k$ converges then so does $\sum_{k} \mathbb{E} X_k^2$.
This is of course plausible but there is one step in the proof that I do not quite understand. Here is the except from the book.
Here, $M_n = \sum_{k=1}^n X_k$, $A_n = \sum_{k=1}^n \mathbb{E} X_k^2$ and $N_n = M_n^2 - A_n$. My question has to do with the use of the stopping time $T$. This is a valid stopping time of course but then Williams derives the inequality $(**)$ assuming that $T$ is finite only to say later that $T$ is inifnite for some $c$ with positive probability. I find the reasoning here very confusing so I would be grateful if someone could give a little more detail. Also, how is it that we have $|M_n^T| \leq K + c$ for every $n$?
Thank you.
The maximum value $|M_n^T|$ can take is given by the "worst-case" where $|M_{n-1}^T| = c$ and $|X_n^T| = K$, which totals $K +c$.
For the second part of your question, let's say $T_c = \inf \{ r \, : \, |M_r| > c\}$. Notice that for any bounded stopping time $\tau$, $|M^\tau - M^{\tau-1}| = |X^\tau| = K$ . In particular $\tau = T_c \land n$ is a bounded stopping time so that inequality holds true. Furthermore, observe that for some $c$, we have that $P(T_c = \infty) > 0$.
Arguing by contradiction, suppose $A_\infty = \sum_k \sigma_k^2 = \infty$. Then, $A_{T_c \land n} 1_{\{T_c = \infty\}} \to A_\infty 1_{\{T_c = \infty\}}$ a.s. and monotonically, so that by the MCT, $$E\left( A_{T_c \land n} 1_{\{T_c = \infty\}} \right) \to E\left (A_\infty 1_{\{T_c = \infty\}} \right) =\infty $$
But this is a contradiction, because by $(\star \star)$ we have $E\left( A_{T_c \land n} 1_{\{T_c = \infty\}}\right) \leq E(A_{T \land n}) \leq (K+c)^2$ for all $n$.