Exercise 1.35
Theorem. The golden ratio \(\varphi\) is a fixed point of the transformation \(x \mapsto 1 + 1/x\).
Proof. We will show that for any \(x > 1\) and \(y = 1 + 1/x\), we have \(|y - \varphi| < |x - \varphi|\), meaning each iteration bring the value closer to \(\varphi\). Recall from [[Exercise 1.13|1.13]] the golden ration equation \(\varphi + 1 = \varphi^2\), or equivalently \(\varphi = 1+1/\varphi\): $$ \begin{aligned} |y - \varphi| &= \left|1 + \frac{1}{x} - \varphi \right| \ &= \left|\frac{x+1-\varphi x}{x} \right| \ &= \left| \frac{x+1-(1=1/\varphi)x}{x} \right| \ \ \ \text{since} \ \varphi = 1 +1/\varphi \ &=\left| \frac{\varphi -x }{x\varphi}\right| \ &= \frac{|x-\varphi|}{|x\varphi|}. \end{aligned} $$ Since \(x>1\) and \(\varphi > 1\), we have \(|x\varphi| > 1\), hence \(|y-\varphi| < |x-\varphi|\) as required. \(\blacksquare\)