Prove $\left(1-\epsilon\right)^x\leqslant\left(1-\epsilon x\right)$ and $\left(1+\epsilon\right)^x\leqslant\left(1+\epsilon x\right)$ Prove the following inequalities with at least two methods: $\forall\, x\in[0, 1]$ and  $\epsilon\in[0, \dfrac{1}{2}]$:
$$
\left(1-\epsilon\right)^x\leqslant\left(1-\epsilon x\right).
$$
$$
\left(1+\epsilon\right)^x\leqslant\left(1+\epsilon x\right).
$$
I would like to get any idea of how to do it (of course you do not need to give me two methods). Any suggestions will be appreciated. 
How I did the first one: I proved that $\left(1-\epsilon\right)^x$ is a convex function of $x$.
How I did the second: I showed also that $\left(1+\epsilon\right)^x$ is a convex function of $x$. So I think I am done.

see the 'generalization' section in here:
http://en.wikipedia.org/wiki/Bernoulli%27s_inequality
