Question 167
Let $a_1, a_2, a_3, \dots$ be an infinite sequence of real numbers such that for each $n \ge 2018$ the number $a_{n+1}$ is the smallest root of the polynomial
$$
P_n(x) = x^{2n} + a_1 x^{2n-2} + a_2 x^{2n-4} + \dots + a_n.
$$
Prove that there exists a positive integer $N$ such that the sequence $a_N, a_{N+1}, a_{N+2}, \dots$ is strictly decreasing.
$$
P_n(x) = x^{2n} + a_1 x^{2n-2} + a_2 x^{2n-4} + \dots + a_n.
$$
Prove that there exists a positive integer $N$ such that the sequence $a_N, a_{N+1}, a_{N+2}, \dots$ is strictly decreasing.