TITLE: Showing that $\sum_{k=N+1}^\infty \frac {1}{k!} < \frac {1}{N!}$
QUESTION [8 upvotes]: I was constructing a proof through inequalities, but I am having a bit of problem showing the following step:  $$\sum_{k=N+1}^\infty \frac {1}{k!} < \frac {1}{N!}$$ 
Is there any quick way to show this?

REPLY [11 votes]: An alternative proof:
$$\sum_{k=N+1}^\infty\frac1{k!}<\sum_{k=N+1}^\infty\frac{k-1}{k!}=\sum_{k=N+1}^\infty\frac1{(k-1)!}-\sum_{k=N+1}^\infty\frac1{k!}=\frac1{N!}$$