Proving that a convex function is Lipschitz

I am trying to show that if $f$ is convex in $(a,b)$ it is Lipschitz in $[c,d]$ where $a \lt c \lt d \lt b$.

Here's what I have so far:

Let $t_1,t_2 \in \mathbb{R}$ such that $a \lt t_2 \lt c \lt d \lt t_1 \lt b$ and let $x_1,x_2 \in [c,d]$.

Because $f$ is convex I know that $$\dfrac{f(c)-f(t_2)}{c-t_2} \lt \dfrac{f(x_2)-f(x_1)}{x_2-x_1} \lt \dfrac{f(t_1)-f(d)}{t_1-d}$$

I think I'm almost there, but what is the right $C$?

Just let $C$ be the max of the absolute value of the functions on the far left and far right of the inequalities, ie:

$$C = \max \left\{\left|\dfrac{f(c)-f(t_2)}{c-t_2}\right|, \left|\dfrac{f(t_1)-f(d)}{t_1-d}\right|\right\}$$