# Prove $e^x - e^y \leq e |x-y|$ for $x$ belonging to $[0,1]$ [closed]

I got as far as saying $e = \frac{e^x - e^y}{x-y}$.
Hint: By MVT , $|e^x-e^y|\leq \sup\limits_{c\in[0,1]}|e^c|\cdot|x-y|$ for $x,y\in[0,1]$ (if your $y$ also in $[0,1]$).