...But I can't get it

We have a sequence of iid nonnegative rv's $\displaystyle (X_n)_{n\in\mathbb{N}}$

Assuming that $\displaystyle E[X_1]$ is finite, we have by the sLLN that $\displaystyle \frac{X_1+\dots+X_n}{n}$ converges to a finite limit.

But how can I conclude that $\displaystyle \lim_{n\to\infty} \frac{X_n}{n}=0$ ?

I tried to relate it to the series $\displaystyle \sum \frac{X_n}{n}$, but the inequality is the other way round than the one that would be helpful...

...so since inequalities don't help me, I hope you guys can help me

Thanks !

Note : I also - miserably - tried Cesàro's mean.