Answer
See below.
Work Step by Step
$a_n$ is a Fibonacci sequence, thus $a_{n+1}=a_n+a_{n-1}$.
Hence: $b_n=\frac{a_{n+1}}{a_n}=\frac{a_n+a_{n-1}}{a_n}=1+\frac{a_{n-1}}{a_n}=1+\frac{1}{\frac{a_n}{a_{n-1}}}$
We know that $\frac{a_n}{a_{n-1}}=b_{n-1}$ by definition. Thus $1+\frac{1}{\frac{a_n}{a_{n-1}}}=1+\frac{1}{b_{n-1}}$. Thus we proved what we had to.
