Is expectation finite almost surely or not almost surely?

Question

Let $\{w_n\}$ be a sequence of non-negative numbers and $\{X_n,\mathcal{F}_n, n\ge 1\}$ be a uniformly bounded martingale differences. Put $M_n= \sum_{k=1}^n w_k^2 <\infty$. I solved that \begin{align*} \displaystyle\sum_{n=2}^\infty \mathbb{E} \left( \left| w_n X_n \right|^2 | \mathcal{F}_{n-1} \right) =\displaystyle\sum_{n=2}^\infty \left[ |w_n|^2 \mathbb{E} \left( \left|X_n \right|^2 | \mathcal{F}_{n-1} \right) \right] &= \displaystyle\sum_{n=2}^\infty \left[ |w_n|^2 \mathbb{E} \left( X_n^2 \right) \right] \text{(by independence)} \\ &\le c^2 \displaystyle\sum_{n=2}^\infty |w_{n}|^2 <\infty \text{ almost surely}. \end{align*} Is this right? Is it finite almost surely or not almost surely?