Prove the upper bound on $T\left(n\right)=T\left(\log_{2}n\right)+O\left(\sqrt{n}\right)$

Question

I need some help with the following recursion:

$T\left(n\right)=T\left(\log_{2}n\right)+O\left(\sqrt{n}\right)$

More specifically I wish to find and prove the upper bound on it.

I have a hunch it is $O\left(\sqrt{n}\right)$ but have gotten stuck proving it really is this.

Any help is appreciated

Thank you!

Answer 1

By simple substitution, you can see that $T(n) = O(\sqrt{n}) + O(\sqrt{\log n}) + O(\sqrt{\log\log n}) + \dots = O(\sqrt{n})$.