Prove by Induction that $r_n$ is $O(\log_2(\log_2n))$ [duplicate]

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• Solving or approximating recurrence relations for sequences of numbers 11 answers

Let the sequence $r$ be defined by:

$$\begin{align*}r_{1} &= 1\\ r_n&= 1 + r_{\lfloor\sqrt{n}\rfloor}\,,\quad n\geq 2\,.\end{align*}$$

Prove by induction that $r_n$ is $O(\log_2(\log_2n))$.

My idea is to find the closed form function of r and then use the big O definition on the closed form function but I'm having a hard time even finding the function in the first place. Any help? Also is this even the right way to approach this proof?

Thanks.