Let $(f_n)$ be the Fibonacci sequence and let $b_n=\dfrac{f_{n}}{f_{n+1}}$. It is easy to show that if $b_n$ converges then it converges to the golden ratio. However, to show that $b_n$ is convergent, we must show that that it is Cauchy. To do so we show that $|b_{n+1}-b_n|<\dfrac{1}{2^n}$ and hence $|b_m-b_n|<\dfrac{1}{2^{m-1}}+\dfrac{1}{2^{m-2}}+\cdots+\dfrac{1}{2^n}$. We then use the geometric sum to find the sum of the RHS.
Is there anyway to show that this sequence converges without using the geometric series here? The constraints are that the problem must be solved using only a knowledge of real sequences (bounded, monotone etc) and not series (which is usually covered after sequences).
