Let $A$ be a non-empty convex subset of $\mathbb{R}^d$. Let $I \in \mathbb{N}^*, x_1, \ldots, x_I \in A^I$ and $(\lambda_1, \ldots, \lambda_I) \in \mathbb{R}_+^I$ such that $\sum_{i=1}^I \lambda_i = 1$, show that:
\begin{itemize}
\item a) $\sum_{i=1}^I \lambda_i x_i \in A$,
\item b) if $x := \sum_{i=1}^I \lambda_i x_i \in \operatorname{Ext}(A)$ then $x_i = x$ for all $i \in \{1, \ldots, I\}$ such that $\lambda_i > 0$.
\end{itemize}