# Proof that the intersection of any finite number of convex sets is a convex set

How to prove that the intersection of any finite number of convex sets is a convex set?

I have no idea.

• Well it makes sense, because it's a going to be a subset of one or more of the convex sets,making it convex..I just don't know how to provide a rigorous proof. – Aggressive Sneeze. Nov 11 '14 at 5:33

Let $(S_i)$ be a convex set for $i = 1,2,\ldots,n$.
For any $x,y \in \cap_{i=1}^n S_i$, $t \in [0, 1]$, we have:
For $i = 1,2,\ldots,n$, $x \in S_i$ and $y \in S_i$ implies $tx + (1-t)y \in S_i$ by convexity of $S_i$.
Hence $tx + (1-t)y \in \cap_{i=1}^nS_i$.
Therefore $\cap_{i=1}^nS_i$ is convex.
• @AggressiveSneeze. No, its valid for all $S_i$ – Aram Nov 11 '14 at 6:34