So I was given a proof: Let $\displaystyle F_n$ be the n-th term of the Fibonacci sequence. Show that $\displaystyle \lim_{n\to\infty}\frac{F_n}{f_{n-1}}=\frac{1+\sqrt{5}}{2}$

I figured out how to do the proof starting with the assumption that the limit exists for $\displaystyle \lim_{n\to\infty}\frac{F_n}{f_{n-1}}$. My problem is that I can't figure out how to show that the limit is convergent. My instructor suggested using something like the squeeze theorem, but I can't figure out how to use the squeeze theorem on a sequence defined recursively.

I'm convinced that the ration decreases to the limit for odd values of n, and increases to the limit for even values of n, but don't know how to prove this, especially without assuming that the limit exists beforehand.

I would appreciate avenues to pursue, but I am trying not to find an already complete proof, so don't give me the answer.

Thanks!