First version of strong law / $\lim a_n^4 = 0 \to \lim a_n = 0$?

Question

From Williams' Probability w/ Martingales:

How does the conclusion follow?

Guess 1:

$E[\sum (\frac{S_n}{n})^4] < \infty$

$\to \sum (\frac{S_n}{n})^4 < \infty$ a.s.

$\to \lim_{n \to \infty} (\frac{S_n}{n})^4 = 0$ a.s.

$\to \lim_{n \to \infty} (\frac{S_n}{n}) = 0$ a.s.?

So $\lim_{n \to \infty} a_n^4 = 0 \to \lim_{n \to \infty} a_n = 0$ ?

Guess 2:

$E[\sum (\frac{S_n}{n})^4] < \infty$

$\to E[\sum (\frac{S_n}{n})] < \infty$

$\to \sum (\frac{S_n}{n}) < \infty$ a.s.

$\to \lim (\frac{S_n}{n}) = 0$ a.s.

Any of those right? If not, how else can I approach this?

Answer 1

Based on Shalop's comments:

Guess 1 is correct because $\lim_{n \to \infty} a_n^4 = (\lim_{n \to \infty} a_n)^4$ because $f(x) = x^4$ is continuous.

Guess 2 is wrong: While $E[X^4] < \infty \to E[|X|] < \infty \to E[X] < \infty$ is true, it is not applicable.

Guess 2 mixes up $\sum (\frac{S_n}{n})^4$ and $(\sum \frac{S_n}{n})^4$