Let $X_n$ be a sequence of uniformly bounded random variables — that is, there exists some $K > 0$ such that $|X_n| \leq K$ almost surely for all $n \in \mathbb N$.

Write $\bar X_N := \frac{1}{N} \sum_{n = 1}^N X_n$, and $Y := \limsup_{N \to \infty} \bar X_N$.

Question: Does there exist some subsequence $X_{n_k}$ such that $\frac{1}{N} \sum_{k=1}^{N} X_{n_k}$ converges to $Y$ almost surely as $N \to \infty$?