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. Justify the existence of a strictly increasing sequence of natural integers $(\phi(j))_{j \in \mathbb{N}}$ satisfying $$\forall j \in \mathbb{N}, \quad \sum_{n > \phi(j)}^{+\infty} a_n^2 \leqslant \frac{1}{8^j}.$$
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. Justify the existence of a strictly increasing sequence of natural integers $(\phi(j))_{j \in \mathbb{N}}$ satisfying
$$\forall j \in \mathbb{N}, \quad \sum_{n > \phi(j)}^{+\infty} a_n^2 \leqslant \frac{1}{8^j}.$$