Prove inquality involving factorial This is part of an analysis problem I'm working on.
Show that:
$ \frac{1}{(n+1)!} + \frac{1}{(n+2)!} + ... < \frac{3}{(n+1)!}$
After some algebra I got
$ \frac{1}{n+2} + \frac{1}{(n+2)(n+3)} + \frac{1}{(n+2)(n+3)(n+4)} + ... < 2$
Now I'm stuck.

Well... you've finished!!!
Assume it's true...
$\frac{1}{(n+1)!} + \frac{1}{(n+2)!} + \cdots < \frac{3}{(n+1)!}$
Now multiplying both side for $(n+1)!$  follows that
$1 + \frac{1}{(n+2)} + \frac{1}{(n+2)(n+3)} + \cdots... < 3$ (1)
Observing that for fixed positive $n$ the inequality $\frac{1}{n + 2 + k} < \frac{1}{2}$ for each $k$ positive the left side of (1) is of course less than
$1 + \frac{1}{(n+2)} + \frac{1}{(n+2)(n+3)} + \cdots... < \sum_{k=0}^{+\infty} \left( \frac{1}{2} \right)^k = 2 < 3$
Then our assumption was correct!!!
