I'm reading Probability Theory, and ran into the following exercise:
Prove that if $X_n \in L^2$ are uncorrelated and identically distributed random variables, then $\frac{1}{n} \sum_{k=1}^n X_k \to \mathbb{E}[X_1]\text{ almost surely.}$
This seems a bit weird, since it is given as a theorem that the convergence happens in probability. (Which implies that I can find a subsequence that converges almost surely.) They don't mention this strengthening of the theorem in the text, so I'm wondering if what the exercise claims is true.