I need to prove that $e^x-e^z+e^y-e^z\geq0$ with $x\leq z\leq y<0.$
Can I do the following:
According to the MVT
$e^x-e^z=\epsilon_1(x-z)$, $e^y-e^z=\epsilon_2(y-z)$ with $\epsilon_1,\epsilon_2<0.$
Can I say that $\epsilon_1<\epsilon_2$ and then
$e^x-e^z+e^y-e^z=\epsilon_1(x-z)+\epsilon_1(x-z)\geq \epsilon_1(x+y-2z)$
and then try to show that $x+y-2z<0$