I was working on a computer program and came up with an intuitive idea that reduces the program module by a considerable length . The idea is intuitive but I never came up with a proof.
Claim : For a natural number n , let S(n) denote the sum of digits of n in its decimal expansion. Prove that there exists a natural number k , such that S(S(S(.......(n))...)) [S composed k times] is a single digit number.
Any help with this proof will be appreciable.