Let $(X_n)_{n \in \mathbb{N}}$ be a sequence of mutually independent random variables satisfying $\mathbb{P}(X_n = -1) = \mathbb{P}(X_n = 1) = \frac{1}{2}$ for all $n \in \mathbb{N}$, and let $(a_n)_{n \in \mathbb{N}}$ be a real sequence such that the series $\sum a_n^2$ converges. For all $N \in \mathbb{N}$, denote $S_N = \sum_{n=0}^N X_n a_n$. Let $(\phi(j))_{j \in \mathbb{N}}$ be a strictly increasing sequence of natural integers satisfying $\sum_{n > \phi(j)}^{+\infty} a_n^2 \leqslant \frac{1}{8^j}$ for all $j \in \mathbb{N}$. Deduce that the event $$\left\{\text{the sequence } \left(S_{\phi(j)}\right)_{j \in \mathbb{N}} \text{ is convergent}\right\}$$ also has probability 1. One may examine the series $\sum \left|S_{\phi(j+1)} - S_{\phi(j)}\right|$.
Let $(X_n)_{n \in \mathbb{N}}$ be a sequence of mutually independent random variables satisfying $\mathbb{P}(X_n = -1) = \mathbb{P}(X_n = 1) = \frac{1}{2}$ for all $n \in \mathbb{N}$, and let $(a_n)_{n \in \mathbb{N}}$ be a real sequence such that the series $\sum a_n^2$ converges. For all $N \in \mathbb{N}$, denote $S_N = \sum_{n=0}^N X_n a_n$. Let $(\phi(j))_{j \in \mathbb{N}}$ be a strictly increasing sequence of natural integers satisfying $\sum_{n > \phi(j)}^{+\infty} a_n^2 \leqslant \frac{1}{8^j}$ for all $j \in \mathbb{N}$. Deduce that the event
$$\left\{\text{the sequence } \left(S_{\phi(j)}\right)_{j \in \mathbb{N}} \text{ is convergent}\right\}$$
also has probability 1. One may examine the series $\sum \left|S_{\phi(j+1)} - S_{\phi(j)}\right|$.