Calculus Level 4

The sequence of functions defined by

$$f_0: x \mapsto \sqrt{x}$$

$$f_{n+1}: x \mapsto \sqrt{x+\sqrt{x-f_n}}$$

converges absolutely in the valid domain and codomain as $$n \to \infty$$

The domain and codomain of the sequence of functions are both $$\mathbb{R}^+$$

Which function does it converge towards?

Bonus: Determine the domain and range of $$f_\infty$$

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