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 an eternal closed loop of 1- 4 - 2 - 1. This has been numerically confirmed for initial numbers up to 260. 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. This problem has been stated by some as perhaps the simplest math problem to state, yet perhaps the most difficult to solve. In this paper, we present a proof that the sequence always converges to 1.