Suppose $\{X_n\}$ is an i.i.d. $\text{Uniform}(0,1)$ sequence of random variables. Define $S_n=X_1+X_2+...+X_n$. Show that $S_n\to\infty$ almost surely.
I believe I have solved the problem and I wish the community would check if what I have done is sufficient, and general, i.e. for any i.i.d. sequence of random variables, my method would work.
We first consider, for $\varepsilon\in(0,1)$, $P(X_n>\varepsilon)=1-\varepsilon$ for all $n\in\mathbb N$ hence $\lim_{n\to\infty}P(X_n>\varepsilon)=1-\varepsilon\neq0$. So $X_n$ does not converge in probability to $0$. So $X_n$ does not converge almost surely to $0$. So $S_n$ diverges almost surely.
EDIT: $S_n$ is an increasing sequence of non-negative terms and we have shown that $S_n$ diverges almost surely. Hence $S_n\to\infty$ almost surely.