Exercise 22, § 4

Let $f_1, \ldots, f_n$ be continuous convex functions in a compact interval $I \subset \mathbf{R}$; suppose that for every $x \in I$, one has $\sup (f_j(x)) \geqslant 0$. Now prove that there exist $n$ numbers $\alpha_j \geqslant 0$ such that $\sum_{j=1}^n \alpha_j = 1$ and that one has $\sum_{j=1}^n \alpha_j f_j(x) \geqslant 0$ in $I$. (First treat the case $n = 2$ by considering a point $x_0$ where the upper envelope $\sup(f_1, f_2)$ attains its minimum; when $x_0$ is interior to $I$, determine $\alpha_1$ in such a way that the left derivative of $\alpha_1 f_1 + (1-\alpha_1) f_2$ at the point $x_0$ is zero. Pass from there to the general case by reasoning by induction on $n$; use the induction hypothesis for the restrictions of $f_1, \ldots, f_{n-1}$ to the compact interval where $f_n(x) \leqslant 0$.)