Proof of the Collatz Conjecture

Take any positive integer N. If it is odd, multiply it by three and add one. If it is even, divide it by two. Repeatedly do the same operations to the results, forming a sequence. It is found that, whatever the initial number we choose, the sequence will eventually descend and reach number 1, where it enters a closed loop of 1- 4 - 2 - 1. This is known as the Collatz conjecture which states that the sequence always converges to 1. So far no proof has ever been found that this holds for every positive integer. In this paper we prove the Collatz conjecture by studying what happens as the sequence becomes infinitely long, thereby avoiding the chaos of finite length sequences. We have noted that the ratio between the number of odd operations and even operations continues to decrease as the sequence length increases, approaching zero for infinite sequence length. This leads to the only possibility that the sequence must be eventually decoupled from its staring value and enter a cycle, with the only possible cycle being the 1-4-2-1 cycle. We have obtained an equation for the final sequence of infinite length, which is the 1-4-2-1 closed loop: