TITLE: Prove that the convex hull of a set is the smallest convex set containing that set
QUESTION [7 upvotes]: How do you prove that the convex hull of A is the smallest convex set containing A?
edit: definition of a convex hull: Given a set A ⊆ ℝn the set of all convex combinations of points from A is called the convex hull of A.

REPLY [3 votes]: Young, when you wrote

How do you prove that the convex hull of A is the smallest set
  containing A?

You meant that convex hull of A is the minimal convex set containing A, right?
To show this, which part is your definition?
The linear-algebraic characterization?
You can see that any intersection of convex sets containing A is also a convex set containing A.