On the Collatz conjecture

In the present work, we suggest a proof for 3n+1 problem which was originally introduced by Lothar Collatz in 1937. Collatz conjecture asserts that the function C: N to N; de�fined by C (n) = 3n + 1 if n is odd positive integer number, and C (n) = n/2 if n is even positive integer number goes to 1. We proof that the k-th iterate of Collatz function Cᵏ (n) is bounded for all positive integer numbers k; n and converges to 1.