How to use this inequality to prove the $\left(1+{1\over n}\right)^n

How to use this inequality to prove the $\left(1+{1\over n}\right)^n

This is an exercise on my text book, given that
if $k>1$, $${n(n-1)\dots(n-k+1)\over k!} \left({1\over
n}\right)^k<{(n+1)n\dots (n-k+2)\over k!} \left({1\over n+1}\right)^k$$
My problem is how to use this inequality to prove that
if $n\geq1$, $$\left(1+{1\over n}\right)^n<\left(1+{1\over
n+1}\right)^{n+1}$$