Convex hull is convex

Question

A set $C \in R^n$ is called convex if the line segment $L = \{ tp + (1-t)p | 0 \lt t \lt 1 \}$ between two arbitrary points $p,p' \in C$ is contained in $C$. The convex hull $CH(C)$ of a set $C \subset R^n$ is the intersection of all convex sets of $R^n$ containing $C$:$$CH(C)= \cap\{C' \subset R^n | C\subset C' , C' convex \} $$ prove that the convex hull is convex.

Answer 1

Pick 2 points in the convex hull. Then by def. Then every C' convex which contains C has the line segment between those points. Thus the intersection of all convex sets containing C has that line segment so the convex hull contains the lime segment