show that if $\beta + n\alpha $ is a root for some integer $n$, then $\beta + n\alpha $ lies in the alpha string through beta. So I would like to show the following, which is,

If $\beta + n\alpha $ is a root for some integer $n$, then $\beta + n\alpha $ lies in the alpha string through $\beta$.

I'm guessing the fact that if $\beta -q\alpha, \ldots , \beta + p\alpha $ is an $\alpha$ string through $\beta$ then $\frac{q-p}{2}\alpha (x) = \beta (x)$ for any $x \in [L_{\alpha},L_{-\alpha}]$.
We have a hint for you:
consider $\alpha$ srting through $\beta$ and $\beta$ string through $\alpha$.
We have a hint for you:
consider $\alpha$ srting through $\beta$ and $\beta$ string through $\alpha$.