e is irrational

We have the series

e^{-1}=\\sum_{k=0}^{\\infty}{(-1)^{k}\\over k!}

Note that this is an alternating series and that the magnitudes of theterms decrease. Hence, for every integer $n>0$ , we have the bound

0<\\left|\\sum_{k=0}^{n}{(-1)^{k}\\over k!}-e^{-1}\\right|<{1\\over(n+1)!},

by the Leibniz’ estimate for alternatingseries (http://planetmath.org/LeibnizEstimateForAlternatingSeries). Assumethat $e=n/m$ , where $m$ and $n$ are integers and $n>0$ . Then we would have

0<\\left|\\sum_{k=0}^{n}{(-1)^{k}\\over k!}-{m\\over n}\\right|<{1\\over(n+1)!}.

Multiplying both sides by $n!$ , this would imply

0<\\left|\\sum_{k=0}^{n}{(-1)^{k}n!\\over k!}-m(n-1)!\\right|<{1\\over n+1},

which is a contradiction because every term in the sum is an integer,but there are no integers between $0$ and $1/(n+1)$ .