BALTIC-WAY 2020 Problem 3

A real sequence $\left(a_{n}\right)_{n=0}^{\infty}$ is defined recursively by $a_{0}=2$ and the recursion formula

$$a_{n}= \begin{cases}a_{n-1}^{2} & \text { if } a_{n-1}<\sqrt{3} \\ \frac{a_{n-1}^{2}}{3} & \text { if } a_{n-1} \geqslant \sqrt{3}\end{cases}$$

Another real sequence $\left(b_{n}\right)_{n=1}^{\infty}$ is defined in terms of the first by the formula

$$b_{n}= \begin{cases}0 & \text { if } a_{n-1}<\sqrt{3} \\ \frac{1}{2^{n}} & \text { if } a_{n-1} \geqslant \sqrt{3}\end{cases}$$

valid for each $n \geqslant 1$. Prove that

$$b_{1}+b_{2}+\cdots+b_{2020}<\frac{2}{3}$$