<s>[INST] Fibonacci Proof Using Induction 
$$f(n) = \left\{\begin{matrix}
0 & n=1\\ 
1 & n=2\\ 
f_{n-1} + f_{n-2} & 
n\geqslant 2\end{matrix}\right.$$

How can I prove by induction that $$f_{n} \leq \left ( \frac{1+\sqrt{5}}{2} \right )^{n-1}$$ for all$$  n\geq l_{a}$$, I have to find the smallest value for $$l_{a}$$
 [/INST] $(1)$ $f_1 \leq (\frac{1+\sqrt{5}}{2})^{1-1} \Rightarrow 0 \leq 1,f_2 = 1 \leq \frac{1+ \sqrt{5}}{2} \approx 1.618 $. $\textbf{True}$
$(2)$ Next: Suppose for some $\textbf{k}$ $\geq n$ we have $f_k \leq (\frac{1+\sqrt{5}}{2})^{k-1}$
$(3)$ We want to show: $f_{k+1} \leq (\frac{1+\sqrt{5}}{2})^{k}$
$(4)$ Well, $f_{k+1} = f_{k-1}+f_{k-2}$
$(5)$ By hypothesis we have bounds on both of the things on right of the inequality.
$(6)$ Thus: $f_{k+1} \leq \ 2(\frac{1+\sqrt{5}}{2})^{k-1} \leq (\frac{1+\sqrt{5}}{2})^{k-1} (\frac{1+\sqrt{5}}{2})^{k} = (\frac{1+\sqrt{5}}{2})^{k}$ 
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