Exercise 17, § 5

Let $g$ be a mapping of ]0, 1] into [−1, 1] such that

Show that there is a continuous increasing mapping $g_2$ and a continuous decreasing mapping $g_1$ of [0, 1] into [−1, 1] such that $g_1(0) = g_2(0) = 0$ and $g_1(x) \leq g(x) \leq g_2(x)$ for $0 < x \leq 1$. [For each integer $n > 0$ consider the greatest lower bound $x_n$ of the numbers $x$ such that $g(x) \geq 1/n$].