1977 Canadian MO Problems/Problem 6

Problem

Let $0<u<1$ and define $u_1=1+u\quad ,\quad u_2=\frac{1}{u_1}+u\quad \ldots\quad u_{n+1}=\frac{1}{u_n}+u\quad ,\quad n\ge 1$ Show that $u_n>1$ for all values of $n=1,2,3\ldots$ .

Solution

Prove by induction that $\forall i>1$ $1<u_i<\frac{1}{1-u}$

$u_1=1+k$ that $1<u_1$ and $1+u<\frac{1}{1-u} \to 1-u^2<1 \to -u^2<0$ .

By induction if $u_i<\frac{1}{1-u}$ then $u_{i+1}=\frac{1}{u_i}+u>(1-u)+u=1 \to u_{i+1}>1$ and if $u_i>1$ then $u_i>1>1-\frac{u^2}{1-u+u^2}=\frac{1-u}{1-u+u^2}$ ( $\frac{u^2}{1-u+u^2}>0$ because $u^2>0$ and $1-u+u^2>1-u>0$ because $1>u$ ).