Suppose $g(x)$ is a convex function (on $\mathbb{R}\to\mathbb{R}$), and $f(x)=jx+k$ (is affine), where $g$ and $f$ intersect on two points $c$ and $d$, $c<d$.
It seems obvious for any points $a < c$ or $a > d$ that $f(d)$ cannot be greater than $g(d)$
But can this be proven using the definition of a convex function?
I think you mean $f(a)$ cannot be greater than $g(a)$.
This is easy. If $a <c$ then we cam write $c=ta+(1-t)d$ for some $t \in (0,1)$. Now $f(c) =tf(a)+(1-t)f(d)=tf(a)+(1-t)g(d)$ and $f(c)=g(c)\leq tg(a)+(1-t)g(d)$. Hence $tf(a)+(1-t)g(d) \leq tg(a)+(1-t)g(d)$ which gives $f(a) \leq g(a)$.