Singleton is Convex Set

Theorem
Let $V$ be a vector space over $\R$ or $\C$, and let $v \in V$.

Then the singleton $S = \set v$ is a convex set.

Proof
For any $x, y \in S$, we have $x = y = v$.

It follows that:


 * $\forall t \in \closedint 0 1: t x + \paren {1 - t} y = v \in S$

Hence $S$ is a convex set.