I am currently working on a book on probability theory and in order to give a proof of Jensen’s inequality for conditional expectancy, the author uses the current property :
Let $g$ be a convex function on $\mathbb R$. Then there exists $2$ sequences of real numbers $(a_n)_{n\in\mathbb N}$ and $(b_n)_{n\in\mathbb N}$ such that $\forall x\in\mathbb R$, we have: $$g(x)=\sup_{n \in \mathbb N} \,(a_nx+b_n)$$
I have never encountered this property before and I’m not sure how to prove it. Despite my quick research, I wasn’t able to find it online. Thanks for the help !
Broad outline: It can be shown that $g$ has left/right derivatives at every point. Consider the lines $y=g(x_0)+g'(x_0+)(x-x_0)$. Convexity implies that this line lies below the graph of $g$ and touches it at $(x_0,g(x_0))$. This gives $g(x)=\sup \{g(x_0)+g'(x_0+)(x-x_0): x_0 \in \mathbb R\}$. We can re-write this as $g(x)=\sup_i (a_ix+b_i)$ for some family $(a_i,b_i), \in I$. Finally we have to reduce this family to a sequence using the fact that any subset of $\mathbb R^{2}$, in particular, $\{(a_i,b_i):i \in I\}$, has a countable dense set.