How to prove Exponential is monotone? How can you prove that if given $x>y$ implies $e^x >e^y$ ?

Hint:
\begin{align} e^x-e^y&=\left(1+x+\frac{1}{2!}x^2+\frac{1}{3!}x^3+\ldots \right) - \left(1+y+\frac{1}{2!}y^2+\frac{1}{3!}y^3+\ldots \right) \\
&=\left(x-y\right)+\frac{1}{2!}\left(x^2-y^2\right)+\frac{1}{3!}\left(x^3-y^3\right)+\ldots  \end{align}
