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I have the following exercise:

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 that if $(f_n)$ is monotone for all n, $(f_n) \to f$ and $f$ is continuous, then $(f_n)$ converges uniformly. By definition of $f_n$, I notice that I only need to prove that $(f_n) \to f$ pointwise. I know that every increasing and bounded sequence converges (Theorem of monotone convergence). My question is, can I use that theorem in a sequence of functions? Is there an easier way to prove the exercise?

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  • $\begingroup$ What you are talking about is Dini's theorem. It requires that $f_n$ be a sequence of continuous functions, so it cannot be used here. $\endgroup$ – user230734 May 24 '15 at 0:16
  • $\begingroup$ No, I am not talkin about Dini's Theorem. $\endgroup$ – jggarita May 24 '15 at 0:19
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    $\begingroup$ I mean the bit with "I know that if $(f_n)$ is monotone for all n, $(f_n) \to f$ and $f$ is continuous, then $(f_n)$ converges uniformly.". That's not true unless the $f_n$'s are continuous, and the space is compact. $[0,R]$ is compact, but continuity isn't achieved. $\endgroup$ – user230734 May 24 '15 at 0:23
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    $\begingroup$ Look at $a_{n+1} = c\cdot a_n + \frac{1}{c\cdot a_n}$. Think Heron. $\endgroup$ – Daniel Fischer May 24 '15 at 0:32
  • $\begingroup$ @BolzWeir In class, we proved that if each of the $f_n$ functions are monotone and $(f_n) \to f$ in a bounded interval and $f$ is continuous, then $(f_n)$ converges uniformly. $\endgroup$ – jggarita May 24 '15 at 18:56

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