# Prove the uniform convergence of $f_{1}(x)= \sqrt x , f_{n+1}(x)=\sqrt{x+f_n(x)}$ in $[0,\infty]$

As far as I understand most of these questions use the M-test, but I can't find a series that suffices.

## 1 Answer

Wait. $\lim\limits_{x\to0}f_n(x)=0$, but $\lim\limits_{x\to0}f_\infty(x)=1$. There is no uniform convergence!

Apparently, $f_\infty(x)=\sqrt{x+{1\over4}}+{1\over2}$

You might want to show the uniform convergence on $(1,\infty)$ or $(\varepsilon,\infty)$; that's another story, and a simple one at that.

• Interesting. maybe the question is flawed. How did you gather that $\lim\limits_{x\to0}f_n(x)=0$ ? – PanthersFan92 Jun 28 '16 at 12:53
• @PanthersFan92 you can do this by induction on $n$, $f_n(0)=0$ and $f_{\infty} (0) =1$ – clark Jun 28 '16 at 12:57