I was researching upon the Collatz conjecture, and I was reading all the research work done by mathematicians including Terry Tao's. I had read that before Terry Tao's research it was proven that almost all Collatz sequences eventually end up below the number you start from. Is it like other sequences which are exceptions are the ones that may grow infinitely?
If proven that all Collatz orbits attain bounded values (hypothetical example), can the statement above be proven for all orbits?



If proven that all Collatz sequences attain bounded values, is it also proven that all sequences end up below the number you start from?
This does not follow. Nothing in that work rules out that there might even be infinitely many distinct cycles from the Collatz function, all spread out a lot. So even if you had that sort of boundedness claim it would not follow.


