A problem related to basic martingale theory

Set $S_n:=\sum_{i=1}^n X_i$ and $S_0=0$. $S$ is a martingale (wrt the natural filtration), so $S^2$ is a sub-martingale. Using the Doob's decomposition we can write $S^2=M+A$ where M is a martingale and A is predictable (increasing) process, both null at 0. It turns out that $A_n=\sum_{i=1}^n \sigma_i^2$.

Define the stopping time $T_\alpha$ as $T_\alpha=\inf\{n \geq0 \ : \ |S_n|>\alpha\}$. By OST we have $$E[S_{T_\alpha \wedge n}^2]-E[A_{T_\alpha \wedge n}]=0;\qquad \forall n.$$

Also $|S_{T_\alpha \wedge n}| \leq \alpha+K$, since $|X_i| \leq K $; then $$E[A_{T_\alpha \wedge n}] \leq (\alpha+K)^2;\qquad \forall n.$$

Since $\sum X_i$ converges a.s., the partial sum $S_n$ are a.s. bounded, then for some $\alpha$ we have that $(T_{\alpha}=\infty)$ has positive probability. So applying this fact to the last inequality, we get $A_{\infty}=\sum\sigma_n^2 < \infty$.

P.S.: I hope it is clear; I am a new user and I want to say hi to everyone. Also I noticed that I can not comment in some post; is there any restriction? Thanks.