Let $X_1, X_2, \ldots$ be random variables such that $P\{X_n \neq 0 \text{ i.o}\} = 1$. From this it follows that $P\{\sum_{k \leq n} |X_k| > 0\} \rightarrow 1$. Show that there exists constants $c_n \in \mathbb{R}$ such that $P\{|c_nX_n| > 1 \text{ i.o}\}=1$.
This seems intuitively clear. Since the $\sum_{k \leq n} |X_k| > 0$ almost everywhere for sufficiently large $n$, it seems like I can find the infimum $\alpha_n = \inf\{X_n(\omega): \sum_{k \leq n} |X_k(\omega)| > 0\}$. Then taking $c_n = 1/\alpha_n$, the value $|c_nX_n| > 1$ almost surely.
Is this the right track? How do I formalize this into a proof?