Rep#2: An algebraic proof of an analytic lemma

There is a rule of thumb that in 90% of all cases when a proof in algebra or combinatorics seems to use analysis, this use can be easily avoided. For example, when a proof of a combinatorial identity uses power series, it is in most cases enough to replace the words ”power series” by ”formal power series”, and there is no need anymore to worry about issues of convergence and well-definedness. When a proof of an algebraic fact works in the field C, it will in most cases work just as well in the algebraic closure of Q, or in any algebraically closed field of characteristic zero, and sometimes even the ”algebraically closed” condition can be lifted, and it is enough to consider a sufficiently large finite algebraic extension of Q. However, as always when it comes to such rules of thumb, there are exceptions. Here is one lemma that is used in various algebraical proofs, and which seems to be really much simpler to prove using analytical properties of C than using pure algebra: