I'm interested in proving part (2) of Theorem 10 here:
Let $\Delta$ be an abstract root system in [a finite dimensional Euclidean space] $V$.
(1) If $\alpha$ and $\beta$ are in $\Delta$, and $\langle \alpha,\beta \rangle>0$, then $\alpha-\beta$ is a root or $0$. If $\langle \alpha,\beta \rangle<0$, then $\alpha+\beta$ is a root or $0$.
(2) If $\alpha \in \Delta$ and $\beta \in \Delta \cup \{0\}$, then the set of elements of $\Delta \cup \{0\}$ of the form $\beta+n \alpha$, $n \in \mathbb{Z}$, contains all and only such elements with $-p \leq n \leq q$, for some $p \geq 0$ and $q \geq 0$ such that $p-q = \frac{2\langle \beta,\alpha \rangle}{\langle \alpha,\alpha \rangle}$.
Hint: To prove (2), assume there is a gap in the set of elements of $\Delta \cup \{0\}$ of the form $\beta + n \alpha$ and use (1) to get a contradiction.
Here is my attempt at part (2):
Let $$S:=\{n \in \mathbb{Z}:\beta + n \alpha \in \Delta\}.$$ As the root system $\Delta$ is finite, so is $S$. Moreover, it is evident that $0 \in S$. Hence we may define $$ -p:=\min S \leq 0, \\ q:=\max S \geq 0. $$
The inner product $$\langle \beta+n \alpha,\alpha \rangle=\langle \beta,\alpha \rangle+n \langle \alpha,\alpha \rangle \tag{$\star$} $$ is positive when $n>-\langle \beta,\alpha \rangle/\langle \alpha,\alpha \rangle$ and negative when $n>-\langle \beta,\alpha \rangle/\langle \alpha,\alpha \rangle$. It could be zero if $n^*:=-\langle \beta,\alpha \rangle/\langle \alpha,\alpha \rangle$ is an integer. Now we prove by cases:
- If $-p=q$ then both must be zero, and the result is obvious.
- If $-p<q$, we can repeatedly apply part (1), starting from both endpoints $n=-p$ and $n=q$ (unless one of them is equal to $n^*$, in which case the other endpoint alone suffices).
In any case, we have shown that $\beta+n \alpha \in \Delta$ for all $-p \leq n \leq q$. The only thing missing is showing that $p-q = \frac{2\langle \beta,\alpha \rangle}{\langle \alpha,\alpha \rangle}$.
Is my proof correct? Any hints on the final part? Thank you!