= Positive roots for a chain Cartan matrix
{title2=$\Phi^+=\{\alpha_i+\cdots+\alpha_j:1\le i\le j\le l\}$}
Suppose the <Cartan matrix> has diagonal two, entries minus one between adjacent indices, and all other entries zero. Normalize the simple-<root of a root system>[root] squared lengths to two. Every <root of a root system>[root] is a <Weyl group> translate of a simple <root of a root system>[root], so all <roots of a root system>[roots] have squared length two. If a positive <root of a root system>[root] has coefficients $c_1,\ldots,c_l$ in the simple-<root of a root system>[root] <basis>, let $c_0=c_{l+1}=0$. Then
$$
2=\sum_{i=0}^l(c_{i+1}-c_i)^2.
$$
The nonnegative integer coefficients force exactly one upward jump of one and one downward jump of one. Thus the coefficients are one on a consecutive interval and zero elsewhere. Conversely, an interval <root of a root system>[root] can be extended by the next simple <root of a root system>[root]: its Cartan pairing with that <root of a root system>[root] is minus one, so the corresponding simple <reflection> adds it. Starting from a simple <root of a root system>[root] constructs every interval and proves the displayed description. This also identifies the <root of a root system>[root] system as type $A_l$ without needing to enumerate <reflection> words individually.
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