Given $n$ iid random variables $X_1, ..., X_n$ with mean $\mu$, then $X_1+...+X_n = n \mu + o(n)$ a.s.

Question

Here's my attempt, is it correct?

Let $S_n = X_1+...+X_n$. Since $S_n / n = \mu$ almost surely as $n \rightarrow \infty$, then $S_n = n \mu + o(n)$ almost surely as $n \rightarrow \infty$.

Answer 1

This is basically correct. However, one would say that $\lim_{n\to\infty}S_n/n=\mu$ almost surely instead of "$S_n / n = \mu$ almost surely as $n \rightarrow \infty$", as equality may not take place.