If $\lim_{n \rightarrow \infty} a_n=L$ then $\lim_{n \rightarrow \infty} f(a_n)=f(L)$?

Question

If we have for example $a_n=1+\sqrt{a_{n-1}}$ and $\lim_{n \rightarrow \infty} a_n=L$ then can I say that $ L=1+\sqrt{L}$? If it's so, what's the proof?

Answer 1

If $f$ is continuous and $a_n$ a converging sequence, then $$f(\lim a_n)=\lim f(a_n)$$ Since $\sqrt{x}$ is continuous, and your sequence bounded, yes, you can say $$\lim a_n=\lim a_{n+1}=\lim (1+\sqrt{a_n})=1+\sqrt{\lim a_n}$$ Thus, $$\lim a_n=\frac 32 + \frac 12\sqrt{5}$$

Answer 2

Yes, yes you can, since for such recursive definitions limits are fixed points of the function relating a_n to a_{n-1} (as long as it is continuous).