Let $\tau$ be a stopping time for the filtration $\{\mathcal{F}_n\}$ and suppose there is a constant N s.t. for every $n\ge 0$, $\mathcal{P}(\tau\le n+N|\mathcal{F}_n)\ge \epsilon \gt 0$ for some $\epsilon$.
Show that $\mathcal{P}(\tau< \infty)=1$ and $\mathbb{E}[\tau^p]< \infty$ for every $p\ge 1$