(related to Corollary: Exponential Function Is Strictly Monotonically Increasing)
Following the order relation for real numbers let \(x < y\). If we set \(\xi=y-x\) then \(\xi > 0\). Therefore, it follows from the definition of the exponential function that
\[\exp(\xi)=1+\xi+\frac {\xi^2}2+\ldots > 1.\]
From the functional equation of exponential function, it follows \[\exp(y)=\exp(x+y-x)=\exp(x+\xi)=\exp(x)\exp(\xi)>\exp(x),\]
which proves that the exponential function is strictly monotonically increasing.