Let $\varphi:[0,+\infty)\to \mathbb{R}$ an increasing continuous function that satisfies that $1/2\leq \varphi(x) <1$ for all $x>0$. Let $f_0:[0,+\infty)\to \mathbb{R}$ an increasing function (not necessarily continuous) such that $f_0(0)>0$. For all $x \in[0,+\infty)$, define: $f_{n+1}(x)=\varphi(x)f_n(x)+\frac{1}{\varphi(x) f_n(x)}$, $n \geq 1$. Show that $(f_n)_{n \in \mathbb{N}}$ converges uniformly in all bounded interval $[0,R]$
I know I am supposed to show my work, but I have no idea on how to solve this problem.
Thanks!