Question 160
Let $a_1, a_2, \dots$ and $b_1, b_2, \dots$ be sequences of real numbers for which $a_1 \gt b_1$ 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 (that is, there exists a positive integer $N$ for which $a_k \lt a_{k+1}$ for all $k \gt N$).
$$
a_{n+1} = a_n^2 - 2b_n
$$
$$
b_{n+1} = b_n^2 - 2a_n
$$
for all positive integers $n$. Prove that $a_1, a_2, \dots$ is eventually increasing (that is, there exists a positive integer $N$ for which $a_k \lt a_{k+1}$ for all $k \gt N$).