Let $\mathrm{\Delta}$ be an indecomposable root system in a real inner product space $E$, and suppose that $\mathrm{\Phi}$ is a simple system of roots in $\mathrm{\Delta}$, with respect to an ordering of $E$. If $\mathrm{\Phi}=\{{\alpha}_{1},\dots ,{\alpha}_{l}\}$, prove that

${\alpha}_{1}+\cdots +{\alpha}_{l}\in \mathrm{\Delta}$

I know that any positive root $\gamma $ may be written as a sum of simple roots, and furthermore that every partial sum is itself a root, but I am unsure if that will help me or not. Any hints to get me started?

${\alpha}_{1}+\cdots +{\alpha}_{l}\in \mathrm{\Delta}$

I know that any positive root $\gamma $ may be written as a sum of simple roots, and furthermore that every partial sum is itself a root, but I am unsure if that will help me or not. Any hints to get me started?