I have seen it stated that Proj of any graded ring $A$, finitely generated as an $A_0$-algebra, is isomorphic to Proj of a graded ring $B$ such that $B_0 = A_0$ and $B$ is generated as a $B_0$-algebra by $B_1$.

Could someone either supply a reference for or a sketch a proof of this statement?

Note: An obvious approach to this question is to make $B$ a Veronese subring of $A$. However, when I try this approach, I end up getting a terrible combinatorics problem that I do not know how to approach.