Use recurrence relations to show that the convergents for $(\sqrt{5}-1)/2$ are ratios of successive Fibonacci numbers.
Here the recurrence relation is: $f_n=0$ if $n=0$, $f_n=1$ if $n=1$, and $f_n=f_{n-1}+f_{n-2}$ if $n\geq2$.
Here the convergents are $c_0=0$, $c_1=1$, $c_2=1/2$, $c_3=2/3$, etc. This pattern is found by looking at the infinite continued fraction of $(\sqrt{5}-1)/2$, which can easily be derived from the infinite continued fraction of the golden ratio.
So far I have tried induction, but got stuck in the middle of my inductive step. I'm not sure if it is a simple algebraic trick I am missing, or if I should approach the proofe from a different angle.
TIA