Show $|\exp(-x/2) - \exp(-y/2)| \leq |x-y|/4$ for $x,y\geq 0$. I am trying to show this inequality:
$$
\left|e^{-x/2} - e^{-y/2}\right| \leq \frac{|x-y|}{4}
$$
for $x,y\geq 0$. 
I've gotten stuck and could use some kind assistance. 
Many thanks in advanced!

Hint: In terms of the distance between $x$ and $y$, it is not hard to show that the maximum difference between $e^{-x/2}$ and $e^{-y/2}$ occurs when either $x$ or $y$ is zero, assuming $x,y \geq 0$. Also, keep in mind what the other answer said about the incorrect constant.
