Nemotron-IMO-Bench 2026 A19

Let $a_1, a_2, \dots$ and $b_1, b_2, \dots$ be sequences of real numbers for which $a_1 > b_1$ and $$a_{n+1}= a_n^2 - 2b_n,\, b_{n+1}= b_n^2 - 2a_n$$ for all positive integers $n$. Prove that $a_1, a_2, \dots$ is eventually increasing, i.e. there exists a positive integer $N$ for which $a_k < a_{k+1}$ for all $k > N$.