A complete proof of convergence for a nearest-prime recurrence to the golden ratio

We prove unconditionally that for any positive integers a1, a2, the sequence defined by a_n = the prime nearest to aₙ₋₁+aₙ₋₂ (n≥3), with ties broken by taking the smaller prime, satisfies limn→∞ a_n/aₙ₋₁ = φ = (1+√5)/2. The proof uses the Baker–Harman–Pintz prime gap theorem (unconditional) and elementary real analysis.