Convexity/The intersection of convex sets is convex

Theorem: Given any collection of convex sets (finite, countable or uncountable), their intersection is itself a convex set.

Proof: If the intersection is empty, or consists of a single point, the theorem is true by definition.

Otherwise, take any two points A, B in the intersection. The line AB joining these points must also lie wholly within each set in the collection, hence must lie wholly within their intersection.