Proposition 19

Let $(\beta_i)_{1 \leq i \leq n}$ be a sequence of positive roots for the chamber $C$ such that $\beta_1 + \beta_2 + \cdots + \beta_n$ is a root. Then there exists a permutation $\pi \in S_n$ such that, for all $i \in \{1, 2, \ldots, n\}$, $\beta_{\pi(1)} + \beta_{\pi(2)} + \cdots + \beta_{\pi(i)}$ is a root.

We argue by induction on $n$, the proposition being clear for $n \leq 2$. Put $\beta = \beta_1 + \cdots + \beta_n$. Then $\sum_{i=1}^n (\beta|\beta_i) = (\beta|\beta) > 0$, so there exists an index $k$ such that $(\beta|\beta_k) > 0$. If $\beta = \beta_k$, then $n = 1$ since $\beta_i > 0$ for all $i$. Otherwise $\beta - \beta_k$ is a root (no. 3, Cor. of Th. 1); it then suffices to apply the induction hypothesis to $\beta - \beta_k = \sum_{i \neq k} \beta_i$.