= Solution
{c}
Put $q=1-\epsilon<1$. The hypothesis implies $\mathbb P(\tau>n+N\mid\mathcal F_n)\leq q$. Since $\{\tau>n\}\in\mathcal F_n$ and survival to $n+N$ implies survival to $n$, the <tower property of conditional expectation> gives
$$
\mathbb P(\tau>n+N)=\mathbb E\left[\mathbf1_{\{\tau>n\}}\mathbb P(\tau>n+N\mid\mathcal F_n)\right]
\leq q\,\mathbb P(\tau>n).
$$
Apply this at $n=0,N,2N,\ldots$ and use $\mathbb P(\tau>0)\leq1$ to obtain
$$
\boxed{\mathbb P(\tau>kN)\leq(1-\epsilon)^k.}
$$
This is a <geometric tail bound from a uniform escape probability>. The <tail-sum formula> for a nonnegative integer-valued variable now gives
$$
\boxed{\mathbb E\tau=\sum_{m\geq0}\mathbb P(\tau>m)
\leq N\sum_{k\geq0}(1-\epsilon)^k=\frac N\epsilon<\infty.}
$$
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