Positive derivative implies increasing

Statement

On an open interval

Suppose $f$ is a function on an open interval $I$ that may be infinite in one or both directions (i..e, $I$ is of the form $\!(a,b)$, $(a,\infty )$, $(-\infty ,b)$, or $(-\infty ,\infty )$). Suppose the derivative of $f$ exists and is positive everywhere on $I$, i.e., $f'(x)>0$ for all $x\in I$. Then, $f$ is an increasing function on $I$, i.e.:

$\forall x_{1},x_{2}\in I,\qquad x_{1}<x_{2}\implies f(x_{1})<f(x_{2})$

Facts used

1. Lagrange mean value theorem

Proof

Fill this in later