Integral Form of Gamma Function equivalent to Euler Form/Lemma

Lemma for Integral Form of Gamma Function equivalent to Euler Form
Let $0 \le t \le m$.

Then:
 * $0 \le e^{-t} - \left({1 - \dfrac t m}\right)^m \le t^2 \dfrac {e^{-t} } m$

Proof
From Exponential of x not less than 1+x:
 * $1 + x \le e^x$

Let $x = \pm \dfrac t m$.

Then:

and:

Then:

Hence: