If $\lim_{n \rightarrow \infty} \frac{S_n^4}{n^4} = 0$ then $\lim_{n \rightarrow \infty} \frac{S_n}{n} = 0$ where $S_n$ is the sum of $n$ iid RVs with mean zero.
My question
I'm having trouble understanding the final part of a proof for the strong LLN. See red box in screenshot below. I'm thinking the explanation is very quick and obvious but ugh I forgot most of HS Calculus. I think I found the proof for the converse here.
Any help is greatly appreciated.
My attempt
If I can say that $\frac{S_n^4}{n^4} > \frac{S_n}{n}$ all the time then if the LHS goes to zero then so must the RHS. But this doesn't seem to be the case always, for example for $n=1$ I don't think I can always say that $X_1^4 > X_1$ since the only thing I know about the $X_i$'s are they have mean zero and they could take fractions as values.
Book proof
