习题 20, § 4

证明凸函数在区间 $I \neq \mathbf{R}$ 上的集合 $\mathscr{K}$ 对于阶是网状的，“$f(x) \leq g(x)$ 对于每个 $x \in I$”（集合理论，III，p. 146）。 Give an example of two convex functions $f, g$ on $I$ such that their infimum in $\mathscr{K}$ takes a value different from $\inf(f(x), g(x))$ at certain points. Give an example of an infinite family $(f_\alpha)$ of functions in $\mathscr{K}$ such that $\inf_{\alpha} f_\alpha(x)$ is finite at every point $x \in I$ and yet there is no function in $\mathscr{K}$ less than all the $f_\alpha$.