How can we prove that slopes increase in a convex function $f: \mathbb{R} \rightarrow \mathbb{R}$ from the definition?
Hint.
Note that if $x\leqslant y\leqslant z$, then we can write $$ y=\lambda x+(1-\lambda)z $$ for some $\lambda\in[0,1]$. Explicitly, you can find what is $\lambda$ in terms of $x,y$ and $z$. On the other hand, the definition of convexity tells you $$ f(y)\leqslant \lambda f(x)+(1-\lambda)f(z). $$ Now do some algebra to get what you want.