Apparently simple inequality on functions in the proof of Bennett's inequality.

Show that
$$
f(u) = u^{-2}(e^u - u- 1) 
$$
is a nondecreasing function of $u$ for $u > 0$.

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My research:

Derivating gives 
$$
f'(u) = \frac{(e^u -1) u^2 - 2u (e^u - u -1)}{u^4} = \frac{e^u (u-2) + (u+2)}{u^3}
$$
Therefore it suffices to show that the denominator is positive.

For $u \ge 2$, $e^u > 1$ implies $e^u(u-2) \ge (u-2)$ which gives the result.

However, for $u < 2$, one might want to derivate again to find the minimum of the denominator, which satisfies $e^u (u-1) + 1 = 0$. It gives an expression of $e^u u $ but I still have to compare $u$ and $e^u$, and can't conclude.