How do I prove Binet's Formula? My initial prompt is as follows:

For $F_{0}=1$, $F_{1}=1$, and for $n\geq 1$, $F_{n+1}=F_{n}+F_{n-1}$.
  Prove for all $n\in \mathbb{N}$:
$$F_{n-1}=\frac{1}{\sqrt{5}}\left(\left(\frac{1+\sqrt5}{2}\right)^n-\left(\frac{1-\sqrt5}{2}\right)^n\right)$$

Which, to my understanding, is Binet's Formula.
I came up with a proof strategy similar to that described here. But as you can see, it is considering the case of $F_n$, not $F_{n-1}$.
So, my question is: does the same strategy hold for $F_{n-1}$? If not, how is it different?

It's impossible to tell without reading your proof, but induction (as a principle) will work regardless of whether you consider $n$ or $n-1$.  The only difference is the indexing of the base case, but with the shift $n \mapsto n-1$ you needn't worry about introducing a "gap" at all.
