Let $(X_n)_{n\in\mathbb{N}}$ be a sequence of i.i.d. random variables with $\mathbb{E}(X_1^+)=\infty$ and $0<c<1$. I'm now trying to prove $$\lim\limits_{n\to\infty}c^n\sum_{k=1}^nX_k=\infty \text{ a.s.} \Longrightarrow \liminf\limits_{n\to\infty}\sum_{k=1}^n c^k X_k>-\infty \text{ a.s.}$$
At first this sounded really trivial, but I don't even have a good starting point to prove it. Could somebody please give me a hint? Thanks in advance!