Bounding the degrees of generators of a homogeneous dimension 2 toric ideal

Let $I$ be the toric ideal defined by a $2\times n$ matrix of integers, $$\mathcal{A}= \left(\begin{array}{cccc} 1&1&\ldots&1\\ a_1&a_2&\ldots&a_n\end{array}\right) $$ with $a_1 < a_2 < \ldots < a_n$. We give a combinatorial proof that I is generated by elements of degree at most the sum of the two largest differences $a_i-a_{i-1}$. The novelty is in the method of proof: the result has already been shown by L'vovsky using cohomological arguments.