Question 64
An infinite increasing sequence $a_{1} \lt a_{2} \lt a_{3} \lt \dots$ of positive integers is called central if for every positive integer $n$, the arithmetic mean of the first $a_{n}$ terms of the sequence is equal to $a_{n}$.
Show that there exists an infinite sequence $b_{1}, b_{2}, b_{3}, \ldots$ of positive integers such that for every central sequence $a_{1}, a_{2}, a_{3}, \ldots$, there are infinitely many positive integers $n$ with $a_{n} = b_{n}$.
Show that there exists an infinite sequence $b_{1}, b_{2}, b_{3}, \ldots$ of positive integers such that for every central sequence $a_{1}, a_{2}, a_{3}, \ldots$, there are infinitely many positive integers $n$ with $a_{n} = b_{n}$.