Exercise 8, § 2

Let E be a hilbertian space which has a countably infinite orthonormal basis $(e_n)_{n \geq 1}$. Let A be the closed convex envelope in E of the set consisting of the points $\left(1 - \frac{1}{n}\right)e_n$ for all $n \geq 1$. Show that there does not exist any pair of points $x, y$ in A whose distance is equal to the diameter of A (compare with IV, p. 54, exerc. 12).