Inequality proof involving series I have a question which starts like this:
Show that for $n>m$ we have $S_m < S_n\leq S_m+\frac{1}{m}\frac{1}{m!}$
Where $S_n=\frac{1}{0!}+\frac{1}{1!}+\frac{1}{2!}+ ... +\frac{1}{n!}$
I have tried using induction on n but that doesn't work for me, can somebody just  point me in the right direction on how I can go about proving this.
thanks

Hint.
First note that the required bound is independent of $n$. So you won't need to use $n$ in an essential way. In the worst case, $n$ will be very large, so you might as well write down the infinite sum $S = \frac{1}{(m+1)!} + \frac{1}{(m+2)!} + \cdots$. You want to prove that we have $S \leq \frac{1}{m} \frac{1}{m!}$. To do this, compare the series $S$ with a convenient geometric series.
