The existence of linear function which smaller than convex functions.

Question

I am trying to prove the Jensen's inequality for conditional expectations.

It is done if we can show that:

With a convex function $\phi$ bounded from below, there exists a linear function $h(x)=ax+b$ such that:

$\phi(x)\geq h(x)$.

So, how to prove the existence of $h(x)$?