Do three points of inequality between convex functions imply inequality over an interval?

Say I have 2 convex functions, $f:\mathbb{R}\rightarrow\mathbb{R}$, and $g:\mathbb{R}\rightarrow\mathbb{R}$. I want to prove that $f(x) > g(x), \forall x \in [l, u]$.

I know that $f(l) > g(l)$ and $f(u) > g(u)$. I evaluate the functions at a third point $l < x_0 < u$, and find that here too $f(x_0) > g(x_0)$. Does this necessarily mean that $f(x) > g(x), \forall x \in [l, u]$?

I cannot find a counterexample, but I also do not know how to prove this property.