EGMO 2026 Problem 4

Let $1 = a_1 \geq a_2 \geq a_3 \geq \cdots$ be an infinite sequence of real numbers such that $a_n = a_{2n} + a_{2n+1}$ for all positive integers $n$. For $r = 2026^{2026}$, prove that

$$\frac{1}{r} \leq a_r \leq \frac{2}{r+1}.$$