Exercise 12, § 7

In the Banach space $l^\infty(\mathbf{N})$, let $e_n$ be the sequence all of whose terms are zero except the n-th which is 1. Let A be the convex closed envelope of the set formed from 0 and the points $e_n/(n + 1)$ ($n \geq 0$). Show that A is compact but that it is not identical with the convex envelope of the set of its extremal points.