# Prove increasing convex function has increasing differences

Let $v: \mathbb{R} \to \mathbb{R}$ be an increasing, convex function.

For any $t>0$ I want to show that for all $x_{1} \leq x_{2}$ we have: $$v(x_{1}+t) - v(x_{1}) \leq v(x_{2} +t) - v(x_{2})$$

This of course can be illustrated heuristically if $v$ is twice differentiable. But I am trying to show this from the definition of a convex function and by the fact that $v$ is increasing, but I am just moving in circles. I am pretty sure this result is true, and I need it to finish a proof I am working on.

Any suggestions will help.

Since $$x_1 \le x_1+t,x_2 \le x_2+t$$, from the definition of convexity we have $$v(x_1+t) \le \left(\frac{x_2-x_1}{x_2-x_1+t}\right)v(x_1) + \left(\frac{t}{x_2-x_1+t}\right)v(x_2+t)$$ and $$v(x_2) \le \left(\frac{t}{x_2-x_1+t}\right)v(x_1) + \left(\frac{x_2-x_1}{x_2-x_1+t}\right)v(x_2+t).$$ Adding up the inequalities gives the desired inequality.
I'm going to assume that $x_2\le x_1+t$. The reasoning in the remaining case is similar. First consider the three points $x_1\le x_2\le x_1+t$. We have $$x_2={x_1+t-x_2\over t}x_1+{x_2-x_1\over t}(x_1+t),$$ so by the convexity inequality for $\nu$, $$\nu(x_2)\le {x_1+t-x_2\over t}\nu(x_1)+{x_2-x_1\over t}\nu(x_1+t).$$ Similarly, $$\nu(x_1+t)\le {x_2-x_1\over t}\nu(x_2)+{x_1-x_2+t\over t}\nu(x_2+t).$$ Now add these two inequalities, clear $t$ from the denominator, and simplify.