Proposition 1

Let $(x_i)$ be a family of points of a convex subset $A$; every barycentre $\sum_i \lambda_i x_i$ of the $x_i$ formed using positive masses $\lambda_i$ (such that $\sum_i \lambda_i = 1$ and $\lambda_i = 0$ except for finitely many of the indices, cf. A, II, § 9.3) belongs to $A$.

Clearly we need only consider the case when the indices are $1, 2, ..., p$ and $\lambda_i > 0$ for each $i$; the proposition is trivial if $p = 1$; we prove the result by induction on $p$. Put $\mu = \sum_{i=1}^{p-1} \lambda_i > 0$, and $y = \sum_{i=1}^{p-1} \frac{\lambda_i}{\mu} x_i$; the induction hypothesis implies that $y \in A$. Now as $\lambda_p = 1 - \mu$ and $\sum_{i=1}^p \lambda_i x_i = \mu y + (1 - \mu) x_p$, its follows from def. 1 that $\sum_{i=1}^p \lambda_i x_i$ belongs to $A$.