Number of times above a linear boundary for a finite variance random walk

Question

I consider a random walk $(S_n)$ with mean zero and finite variance, and $\epsilon>0$. Is it true that

$$ \mathbb{E}\left[\sum_{n=0}^{+\infty} 1_{S_n>n\epsilon}\right] < +\infty \quad ? $$

If no, is it true if we add the hypothesis $\mathbb{E}\left(|S_1|^{2+\delta}\right)<+\infty$ ?