Consider a process $\{ X_n ;n\in \mathbb{N}_0 \}$, in a Euclidean space. Let a Borel set $S$ be given, and define $T = \inf\{ i \in \mathbb{N}_0: X_i \notin S\}$ with the natural filtration provided.
Suppose $\exists \epsilon >0$ and $\exists N > 0$ such that for all $n \in \mathbb{N}_0$ and $x \in S$
$$P(X_{n+N} \notin S|\mathcal{F}_n) \geq \epsilon 1_{X_n \in S}$$ Show that $E[T^k] < \infty$ for any $k \in \mathbb{N}$.
Been stuck on this for a couple of days now and don't know how to proceed.