Remark

When the set $H$ satisfies the conditions of Proposition 2, it is a lattice with respect to the ordering $f \leq g$. But it should be remarked that a subset $H$ of $C$ can be a lattice with respect to this ordering without it being necessarily the case that the least upper bound (resp. greatest lower bound) in $H$ of two functions $u, v$ of $H$ is the same as their least upper bound (resp. greatest lower bound) in $C$. \* An example is provided by the convex mappings of a compact interval of $\mathbf{R}$ into $\mathbf{R}$. \*