At the begining of the Chapter 21 of Peter Lax's functional analysis, he says that
If C is a precompact set in a Banach space, so is its convex hull.
can be deduced from the following properties:
(a) S is precompact if and only if every sequence of points of S contains a Cauchy subsequence.
(b) S is precompact iff for every $\epsilon >0 $ it can be covered by a finite number of balls of radius $\epsilon$.
I can not see how we can deduce it from these properties, can someone help me? Thanks!