NL-NWO 1973 Problem 5

An infinite sequence of integers $a_1,a_2,a_3, ...$ is given with $a_1 = 0$ and further holds for every natural number $n$ that $a_{n+1} = a_n - n$ if $a_n \ge n$ and $a_{n+1} = a_n + n$ if $a_n < n$ . (a) Prove that there are infinitely many numbers in the sequence equal to $0$ . (b) Express in terms of $k$ the ordinal number of the $k^e$ number from the sequence, which is equal to $0$ .