A general convergence theorem for nearest-element recurrences

Abstract

Let b(n) be a strictly increasing sequence of positive integers satisfyinglim_{n→∞} b(n+1)/b(n) = 1. Define a recurrence by taking a_n to be the element of b nearest to a_{n-1} + a_{n-2}, with ties broken by taking the smaller element. We prove that for any initial values a_1, a_2 in b, lim_{n→∞} a_n / a_{n-1} = φ := (1 + √5)/2. This generalizes the case b(n) = p_n (the primes) studied in OEIS A394720. The proof uses only elementary real analysis and the density condition b(n+1)/b(n) → 1; no unproved conjectures are needed. The author used a large language model for language polishing and formatting only. All mathematical content was independently developed and verified by the author.