Binet's formula for the Fibonacci numbers

1. The roots of $r^2-r-1=0$ are $\varphi=\frac{1+\sqrt5}2$ and $\psi=\frac{1-\sqrt5}2$. So $F_n=A\varphi^n+B\psi^n$. From $F_0=0$: $B=-A$. From $F_1=1$: $A(\varphi-\psi)=A\sqrt5=1$. Hence
$$F_n=\frac{\varphi^n-\psi^n}{\sqrt5}.$$

2. Since $|\psi|<1<\varphi$, we have $\left(\frac\psi\varphi\right)^n\to0$ and
Dividing numerator and denominator by $\varphi^n$:
$$\frac{F_{n+1}}{F_n} = \frac{\varphi^{n+1}-\psi^{n+1}}{\varphi^n-\psi^n}=\frac{\varphi-\psi\left(\frac\psi\varphi\right)^n}{1-\left(\frac\psi\varphi\right)^n}\longrightarrow\varphi.$$