Show that $\mathcal{C}(A)$ is the smallest convex of $E$ containing $A$.

Question

Let $E$ be a $\mathbb{R}$-vector space, and $A$ a nonempty subset of $E$.

Show that $$\mathcal{C}(A) = \biggl\{\sum \limits_{k=1}^n \lambda_kx_k \biggm| n \in \mathbb{N}^*, (x_1,\dots,x_n) \in A^n, (\lambda_1,\dots,\lambda_n) \in \mathbb{R}_+^n \text{ and } \sum \limits_{k=1}^n \lambda_k = 1\biggr\}$$ is the smallest convex of $E$ containing $A$.

My attempt ;

1) $\mathcal{C}(A)$ is convex: Clear using $\phi(t) = xt+(1-t)y$ and writing $x,y$ as defined.

2) It contains $A$: Just take $n=1$

For the smallest case I'm stumped.

Answer 1

Hint: prove that, if $B$ is a convex set in $E$, $x_1,x_2,\dots,x_n\in B$, $\lambda_1,\dots,\lambda_n\in\mathbb{R}_+$ and $\lambda_1+\dots+\lambda_n=1$, then $$ \sum_{k=1}^n\lambda_kx_k\in B $$ (induction on $n$).