Intersections of convex sets are convex

Proposition. Let $\{\Omega_i\}_{i\in I}$ be a family of convex subsets of a vector space $X$. Then the intersection $\bigcap_{i\in I}\Omega_i$ is convex.

Context. This is fundamental for defining the convex hull as an intersection of all convex supersets and for building convex feasible regions from many convex constraints.

Proof sketch. If $x,y$ lie in every $\Omega_i$, then for each $i$ and each $\lambda\in[0,1]$ the point $\lambda x+(1-\lambda)y$ lies in $\Omega_i$ by convexity. Hence it lies in the intersection.