<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] $f_1 = 0 \leq \left(\frac{1+\sqrt{5}}{2}\right)^0 = 1$
$f_2 = 1 \leq \frac{1+ \sqrt{5}}{2} \approx 1.618$
Suppose $f_k \leq \left(\frac{1+\sqrt{5}}{2}\right)^{k-1}, f_{k+1} \leq \left( \frac{1 + \sqrt{5}}{2} \right)^{k}$
Then $f_{k+2} = f_k + f_{k+1} \leq \left(\frac{1+\sqrt{5}}{2}\right)^{k-1} + \left( \frac{1 + \sqrt{5}}{2} \right)^{k} = \left(\frac{1 + \sqrt{5}}{2}\right)^{k+1}$
Proof of last statement: $a^{k+1} = a^k + a^{k-1} \rightarrow a^2 - a - 1 \rightarrow a = \frac{1 + \sqrt{5}}{2}$
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