 Solution

Let $\beta$ be a positive root. If it is not simple, it is a sum of two positive roots. Repeating this decomposition terminates because the height with respect to a regular positive functional strictly decreases, and it writeswith at least one $n_i>0$. Equivalently, one may repeatedly choose a simple root ${\alpha }_i$ with $(\beta ,{\alpha }_i)>0$ and use the root string to replace $\beta$ by the root $\beta -{\alpha }_i$.

For uniqueness, the simple roots are linearly independent. Therefore two such expansions have identical coefficients. Here “positive integer coefficients” must allow zero coefficients: a simple root itself has coefficient one on its own basis vector and zero on the others.