Union of Chain of Convex Sets in Vector Space is Convex
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Theorem
Let $\GF \in \set {\R, \C}$.
Let $X$ be a vector space over $\GF$.
Let $\Gamma$ be a chain of convex sets.
Let:
- $\ds C = \bigcup \Gamma$
Then $C$ is convex.
Proof
Let $t \in \closedint 0 1$ and $x, y \in C$.
Then there exists $C_1, C_2 \in \Gamma$ such that $x \in C_1$ and $y \in C_2$.
Since $\Gamma$ is a chain, we have $C_1 \subseteq C_2$ or $C_2 \subseteq C_1$.
Without loss of generality suppose that $C_1 \subseteq C_2$.
Then $x, y \in C_2$.
Since $C_2$ is convex, we have $t x + \paren {1 - t} y \in C_2$.
So $t x + \paren {1 - t} y \in \bigcup \Gamma = C$.
So $C$ is convex.
$\blacksquare$