I am having trouble understanding the proof for:
4.3.1 Lemma. Convex hull $C$ of a set $X \subseteq \Re $ equals the set: $D= \left\{ \sum_{i=1}^{m}{t_i x_i} : m \geq 1, x_1...x_m \in X, t_1,...,t_m \geq 0, \sum_{t=1}^{m}{t_i} = 1 \right\}$ of all convex combinations of finitely many points of X.
For the direction $C\subseteq D$, the authors say:
For the reverse inclusion it suffices to prove that $D$ is convex, that is, to verify that whenever $x,y \in D$ are two convex combinations and $t\in (0,1)$, then $tx + (1-t)y$ is again a convex combination.
Why does it suffice to prove that $D$ is convex? Shouldn't we prove that any point in $C$ is in $D$?
P.S. And what would it mean for a point to be in $C$ - which is convex hull? I find that confusing...