Show that $\limsup_{n\rightarrow\infty} \frac{\sum_{k=1}^{n} X_k}{n}<\infty$ a.s.

Question

Let $X_k, k \geq 1$, be i.i.d. random variables such that $\limsup_{n\rightarrow\infty} \frac{X_n}{n}<\infty$ a.s, then show that $\limsup_{n\rightarrow\infty} \frac{\sum_{k=1}^{n} X_k}{n}<\infty$ a.s.

I'm thinking to use Borel-Cantelli lemma but don't know where to start. Any hints would be helpful. Thanks.

Answer 1

I think the Stolz-Cesaro's lemma will solve the problem. https://en.wikipedia.org/wiki/Stolz%E2%80%93Ces%C3%A0ro_theorem