I've been asked to show that, for a sequence of independent random variables $(X_n)$, if
$$\mathbb{P}[\lim_{n\rightarrow \infty} \frac{1}{n}\sum_{k=1}^{n} X_k = 1] > 0$$
then $$\frac{1}{n}\sum_{k=1}^{n} X_k \rightarrow 1$$ almost surely.
I'm really not sure how to begin. I thought it might be possible with the Borel Cantelli lemma, but the sequence of sums is not an independent sequence, which is required in the statement. Any help would be appreciated.