Given a function $f_{1}(x)=\sqrt{x}, f_{n+1}(x)=\sqrt{x+f_{n}(x)}$ for $x \in[0, \infty)$. How can I prove if $f_{n} \rightarrow f$ uniformly or pointwise ? I tried with the quotient criteria to prove that $f_{n}$ converges. However $\frac{f_{n+1}}{f_{n}}>1$. Also how can I determine $f'$ or $f'_{n}$?
A rather theoretical question if $f_{n} \rightarrow f$ will $f'_{n} \rightarrow f'$? Keywords regarding the topics for further reading are also appreciated.