Q: Let $\displaystyle \alpha$ be any real number larger than $\displaystyle \beta = (1/2)*(1 + \sqrt{5}) = 1.61 . . . .$

Prove by induction or otherwise that for $\displaystyle n \geq 1, F_n < \alpha^2$

(Note that $\displaystyle \beta$ is a solution of the equation $\displaystyle x^2 = x+1$)

What I have so far:

Base Case:

n = 1

$\displaystyle \alpha > \beta > F_1 = 1$

$\displaystyle \Rightarrow F_1 < \alpha$

Inductive Step

Assume $\displaystyle F_k < \alpha^k$ for n = k

$\displaystyle F_{k+1} = F_k + F_{k-1} < F_k + F_k = 2*\alpha^k < \beta^2*\alpha^k = (\beta + 1)*\alpha^k < \alpha^2*\alpha^k = \alpha^{k+2}$

I've had other attempts but this is the closest I could come up with.