Proposition 3

Let $(f_\alpha)$ be a family of convex functions on an interval $I \subset \mathbf{R}$; if the upper envelope $g$ of this family is finite at every point of $I$ then $g$ is convex on $I$.

Indeed, the set of points $(x, y) \in \mathbf{R}^2$ lying above the graph of $g$ is the intersection of the convex sets formed by the points lying above the graph of each of the functions $f_\alpha$; so it is convex.