Dilation of Convex Set containing Zero Vector by Real Number between 0 and 1

Theorem

Let $\GF \in \set {\R, \C}$.

Let $X$ be a vector space over $\GF$.

Let $C$ be a convex set with ${\mathbf 0}_X \in C$.

Let $t \in \closedint 0 1$.

Then:

$t C \subseteq C$

Proof

Let $x \in t C$.

Then we have $x \in t C + \paren {1 - t} {\mathbf 0}_X$.

Since ${\mathbf 0}_X \in C$, we have $x \in t C + \paren {1 - t} C$.

By definition 2 of a convex set, we have $t C + \paren {1 - t} C \subseteq C$.

So we have $x \in C$.

So $x \in t C$ implies that $x \in C$.

So $t C \subseteq C$.

$\blacksquare$