In Bellman's Dynamic Programming (1957), regarding the equation,
$$ f(x) = \max_{0 \leq y \leq x} \left[ g(y) + h(x-y) + f(ay+b(x-y)) \right]$$
He writes [about extending this to an infinite recursion],
... we may encounter many of the usual difficulties associated with infinite processes. It is, first of all, no longer clear that a maximum exists rather than a supremum. That is to say, there may be no allocation policy which actually yields the total return $f(x)$.
I understand the definition of supremum requires two sets, $S$ and $X$, $S \subset X$. My question is, what are two sets in this particular context. More generally, is there a convention for when the sets are not mentioned explicitly?
The question is answered in Dynamic Programming and Optimal Control (Bertsekas, 2017):
$S = \left\{ f(x) | y \in \left[ 0, x \right] \right\}$
$X = \mathbb{R}$