Question 55
Given a positive integer number $k$, define the function $f$ on the set of all positive integer numbers to itself by
$$
f(n) = \begin{cases} 1, & \text{if } n \le k+1, \\ f(f(n-1)) + f(n - f(n-1)), & \text{if } n \gt k+1. \end{cases}
$$
Show that preimage of every positive integer number under $f$ is a finite non-empty set of consecutive positive integer numbers.
$$
f(n) = \begin{cases} 1, & \text{if } n \le k+1, \\ f(f(n-1)) + f(n - f(n-1)), & \text{if } n \gt k+1. \end{cases}
$$
Show that preimage of every positive integer number under $f$ is a finite non-empty set of consecutive positive integer numbers.