Exercise 2, § 4

Show that every convex function $f$ on a compact interval $I \subset \mathbf{R}$ is the limit of a decreasing uniformly convergent sequence of convex functions on $I$ which admit a second derivative on $I$ (first consider the function $(x-a)^+$, and approximate $f$ by the sum of an affine linear function and a linear combination $\sum_j c_j (x-a_j)^+$ with coefficients $c_j \geq 0$).