ATRACTOR

Note that \(f_{3}=2>\frac{8}{5}=1,6\). Suppose that the inequality is true for the natural numbers from \(3\) to \(m\). Then \[\begin{array}{cl} f_{m+1} & =f_{m}+f_{m-1}>\left(\frac{8}{5}\right)^{m-2}+\left(\frac{8}{5}\right)^{m-3}=\\ & =\left(\frac{8}{5}\right)^{m-1}\left[\frac{5}{8}+\left(\frac{5}{8}\right)^{2}\right]=\\ & =\left(\frac{8}{5}\right)^{m-1}\left(\frac{65}{64}\right)>\left(\frac{8}{5}\right)^{m-1}. \end{array}\]