Convex hull is the smallest convex set containing Ω

Proposition. For any set $\Omega\subset X$ in a real vector space, the convex hull $\mathrm{co}(\Omega)$ satisfies:

1. $\mathrm{co}(\Omega)$ is convex.
2. $\Omega\subset \mathrm{co}(\Omega)$.
3. If $C$ is convex and $\Omega\subset C$, then $\mathrm{co}(\Omega)\subset C$.

Context. This states precisely that $\mathrm{co}(\Omega)$ is the minimal convex superset of $\Omega$.

Proof sketch. By construction, $\mathrm{co}(\Omega)$ is an intersection of convex sets containing $\Omega$, so it contains $\Omega$ and is contained in each such set. Convexity follows from closure of convexity under intersections.