Let $A,B$ be two countable sets of real functions ($\mathbb R\to\mathbb R$), satisfy the property that for all $f\in A,g\in B$, there is $G\in\mathbb R$, such that $\forall x>G,f(x)\le g(x)$.
Prove that there is a real function $h:\mathbb R\to\mathbb R$ st. $\forall f\in A,g\in B,\exists G\in\mathbb R,\forall x>G,f(x)\le h(x)\le g(x)$.
Easy to see that if $A$ or $B$ is uncountable the result will fail.
Moreover, define a preorder between real functions, say $f\le g$, if $\exists G$, st $\forall x>G,f(x)\le g(x)$. It raises to a partial order if we make equivalent classes in the set of real functions.
That is same to say if $A\le B$ then there is a $h$ st. $A\le h\le B$.
What kind of structure does this partial set has?
Let $(X,\leq)$ denote the poset of equivalence classes $[f]$ of maps $f \in \mathbb{R}^{\mathbb{R}}$, then for $[f],[g] \in X, [f] \vee [g] := [\max(f,g)]$ and $[f] \wedge [g]:= [\min(f,g)]$ are well defined and are join and a meet operations for $(X,\leq)$ which is thus a lattice. Moreover $(X,\leq)$ and $(X,\geq)$ are isomorphic via the involution $[f] \mapsto [-f]$.
What you have to prove can be deduced from the fact that any countable subset of $X$ has a join. Do you know how to prove that?
Some properties of $(X,\leq)$, like the least cardinal of an unbounded subset of $X$, might depend on properties of the continuum.