I am trying to understand the part of the proof In Fulton's and Harris's Represantation Theory book where he shows that the length of the root is at most 4:
Theorem If $\alpha,\beta$ are roots with $\beta \neq \pm \alpha$, then the $\alpha$ string through $\beta$, i.e. the roots of the form \begin{equation*} \beta - p\alpha, \beta -(p-1)\alpha, \dots, \beta - \alpha, \beta, \beta + \alpha, \beta + 2\alpha, \dots, \beta + q\alpha \end{equation*} has at most four in a string, i.e. $p+q \leq 3$; in addition, $p-q=n_{\beta\alpha}$
where $n_{\beta \alpha}= 2\frac{(\beta,\alpha)}{(\alpha,\alpha)}$ (positive definite form on a vector space $V$) and then the reflection of a root $\beta$ in the hyperplane orthogonal to a root $\alpha$ is given by \begin{equation*} W_{\alpha}(\beta) = \beta - n_{\beta \alpha}\alpha \end{equation*}
The part of the proof I don't understand is when he writes that \begin{equation} W_{\alpha}(\beta + q\alpha) = \beta - p\alpha \quad (1) \end{equation} , meaning that the reflection (in the hyperplane orthogonal to $\alpha$) reverses a root string. I read somewhere else that this is geometrically an obvious fact but I can't see how. I get that by reflecting through $\alpha$ we just add a positive or negative multiple of $\alpha$ to any root but I really cannot see why $(1)$ is true.
It follows from the definition of $W_\alpha$ that for $k \in \mathbb Z$, $W_\alpha(\beta+k\alpha)$ is of the form $\beta + \ell(k) \alpha$ for some $\ell(k) \in \mathbb Z$. Consequently, that root string is invariant under $W_\alpha$. Further, because $W_\alpha(\alpha) = -\alpha$ and $W_\alpha$ is linear, we have
$$k_1 > k_2 \Rightarrow \ell(k_2) > \ell(k_1),$$
i.e. $\ell$ reverses the order of the root chain. That's enough to prove the statement $W_\alpha(\beta +q\alpha) =\beta-p\alpha$, since these are the two "end points" of the chain.