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.

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.
