Given a positive integer number k, define the function f on the set of all positive integer numbers to itself by f(n)={1,if n≤k+1f(f(n−1))+f(n−f(n−1)),if n>k+1 Show that the preimage of every positive integer number under f is a finite non-empty set of consecutive positive integers.
Problem
Source: American Mathematical Monthly
Tags: function, induction, strong induction, algebra proposed, algebra