Convex Hull is Smallest Convex Set containing Set

Theorem
Let $X$ be a vector space.

Let $U \subseteq X$.

Let $\map {\operatorname {conv} } U$ be the convex hull.

Then $\map {\operatorname {conv} } U$ is the smallest convex subset of $X$ containing $U$ in the sense that:


 * $\map {\operatorname {conv} } U$ is convex and if $K \subseteq X$ is a convex subset with $U \subseteq K$, we have that $\map {\operatorname {conv} } U \subseteq K$.

Proof
First we prove that $\map {\operatorname {conv} } U$ is convex.