Suppose $(\Omega, \mathcal{F}, \mathbb{P})$ is a probability space. Suppose $X_1, X_2, \ldots$ are i.i.d. random variables. For $S_n = X_1 + \cdots + X_n$, assume
$\frac{S_n}{n}\rightarrow 0$ in measure
$\frac{S_{2^n}}{2^n}\rightarrow 0$ in almost surely.
Show that $\frac{S_n}{n}\rightarrow 0$ almost surely.
My attempt: Note
$$\sum_n E\biggl[\biggl(\frac{S_n}{n}\biggr)^2\biggr] = E\biggl[\sum_n \frac{(S_n)^2}{n^2} \biggr]$$
Hence the sum is equal to
$$\sum_{n=1}^{\infty}\frac{1}{n^2} E\biggl[ \sum_{i=j}^nX_i^2+2\sum_{i\neq j}^nX_iX_j \biggr], $$
and this is equal to
$$ \sum_{n=1}^{\infty}\frac{1}{n}{E[X_1^2]}+\frac{2}{n}E[X_1]^2 $$
But I'm not sure what else to do (also any advice would certainly be appreciated).
How should I approach this?