The following is an application of the Wald's equation, i.e., if $X_n$ are iid with $E|X_n|<\infty$ and $N$ is a stopping time with $EN<\infty$, then $ES_N=EX_1 EN$. The key point of the proof is showing that $EN<\infty$, but I don't see how to get this from $P(N>n(b-a))$ is summable over $n$. I know that $EN=\sum_{n=1}^\infty P(N>n)$, but this isn't equal to $\sum_{n=1}^\infty P(N>n(b-a))$ is it? I would greatly appreciate any help.
If $N$ is a nonnegative random variable and $x > 0$, then the summability of $P(N > nx)$ is equivalent to the summability of $P(N/x > n)$. But this means that $N/x$ has a finite expectation, which implies that $N$ must also have a finite expectation.