>To show what we want, we must prove the stronger condition that F(n)φ - F(n+1) → 0. Noting that {F(n+1)/F(n)} is the sequence of convergents of the continued

>To show what we want, we must prove the stronger condition that F(n)φ - F(n+1) → 0. Noting that {F(n+1)/F(n)} is the sequence of convergents of the continued fraction expansion of φ, your stronger condition follows from Theorem 5 at https://en.wikipedia.org/wiki/Continued_fraction https://en.wikipedia.org/wiki/Continued_fraction .