Does the Strong Law of Large Numbers imply the following?

Question

The Strong Law of Large Numbers in my Probability textbook is given as follows. Let $X_n$ be a sequence of identically distributed pairwise independent $\mathbb{R}$-valued random variables. Let $S_n$ denote the corresponding sequence of partial sums. Then as $n\rightarrow\infty$ we have $\frac{S_n}{n}\rightarrow E(X_1)$ almost surely.

I was wondering about the following. Suppose $E(X_1)>0$. Seems like a natural implication of the SLLN to conclude that $S_n$ and $n E(X_1)$ converge to the same limit almost surely. This limit for $n E(X_1)$ is $\infty$, so $S_n\rightarrow\infty$ almost surely. Is my reasoning correct?

Answer 1

In order to make the "Seems like" more rigorous, one could write that for almost every $\omega$, there exists an integer $N$ depending on $\omega$ such that if $n \geqslant N$, then by the strong law of large numbers, $S_n(\omega)/n\gt \mathbb E[X_1]/2 $, hence for such $n$'s, $$S_n(\omega)\geqslant n\frac{\mathbb E[X_1]}2 .$$