I've been thinking of something like this: if you can prove there is no other recurring pattern (of all kinds), it's obvious that any sequence ends up with 4,2,1 (eventually it will get to 4 and then it's stucked).

It's easy to prove there are no others positive recurring patterns length of 2,3,4 end even 5 numbers. I think that if you continue with this line you will maybe find the desirable proof :)