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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 $x \in [L_{\alpha},L_{-\alpha}]$.

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I'm confused about what definitions you are using. To me, the $\alpha$ string through $\beta$ consists of roots of the form $\beta + n\alpha$ for some integer $n$. – Sammy Black May 30 '13 at 21:40
The alpha string through beta consists of a sequence of weights. But maybe there is a break in the sequence. And so there is more than just one string. The point of this is to show there is only one string. – user58514 May 30 '13 at 21:43

We have a hint for you: consider $\alpha$ srting through $\beta$ and $\beta$ string through $\alpha$.

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