As far as I understand most of these questions use the M-test, but I can't find a series that suffices.
2026-04-21 21:31:28.1776807088
Prove the uniform convergence of $f_{1}(x)= \sqrt x , f_{n+1}(x)=\sqrt{x+f_n(x)}$ in $[0,\infty]$
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Wait. $\lim\limits_{x\to0}f_n(x)=0$, but $\lim\limits_{x\to0}f_\infty(x)=1$. There is no uniform convergence!
You might want to show the uniform convergence on $(1,\infty)$ or $(\varepsilon,\infty)$; that's another story, and a simple one at that.