Prove that for all x > 0 and all positive integers n

$\displaystyle e^x > 1 + x + x^2/2! + x^3/3!+ ... + x^n/n!$

Hint:

$\displaystyle e^x $= 1 + $\displaystyle \int _0\,^x\!e^tdt $ > 1 + $\displaystyle \int _0\,^x\!dt $ = 1+ x

$\displaystyle e^x $= 1 + $\displaystyle \int _0\,^x\!e^tdt $ > 1 + $\displaystyle \int _0\,^x\!(1+t)dt $ = $\displaystyle 1+ x + x^2/2$, and so on.

Prove by induction on n >= 1, must jutify the hint