calculus - Why is $sumlimits_{n=N+1}^{infty} frac {1}{2^n!} < sumlimits_{n=(N+1)!}^{infty} frac {1}{2^n} = frac{1}{2^{(N+1)!-1}}$

I'm reading a proof of a theorem and some applications and in the process the author used the following without any explanation of why it's true. I'm trying to understand why exactly this is true, but I have no idea how one would justify it besides saying it seems true intuitively. Any help would be much appreciated.

$$
\sum_{n=N+1}^{\infty} \frac {1}{2^n!} < \sum_{n=(N+1)!}^{\infty} \frac {1}{2^n} = \frac{1}{2^{(N+1)!-1}}
$$