That part of the "proof" is actually correct. If x is odd it can be written as 2y+1 (for some integer y) so 3x+1 becomes 3(2y+1)+1 = 6y+4 = 2(3y+2), which is obviously even.
The final line of the proof is wrong. Firstly proving things about how things work on average is not the same as proving that something can never happen. For example if I continually flip a coin the percentage of flips that land on heads will converge to 50%, that doesn't mean it is impossible for me to flip all tails (and get an observed percentage of 0%). Secondly even if at each stage you multiply by 1.5 or 0.5 with equal probability, the expected value doesn't go down to 0, it actually stays the same as what you started with (although in lots of cases you will end up slightly below the original number you started with, in a few cases will end up with a number much bigger than what you started with, so they balance out).
Anything to do with the collatz conjecture, riemann hypothesis, anything by David Cole (aka primework123), and any posts claiming to have discovered an amazing new result (or disproof of an existing result) should be ignored. No legitimate amazing new discovery will ever appear on this forum or similar forums. The result should be ignored, unless it appears in a reputable peer reviewed journal (there are plenty of non-peer reviewed journals which will print anything and should be ignored) or has the legitimate backing of a reputable researcher employed in a reputable well known university (and never take the poster's word that this is the case).
Check outhttps://www.quora.com/Why-do-math-crank ... rish-exist
and David Joyce's answer (which explains that some people are just misguided, whilst others are blind to their mistakes)
It is best to ignore posts like these, rather than engage the author's in a pointless discussion about why they're wrong, neither side will convince the other. If I'm totally honest, the site admin (aka Math Tutor) should be removing these posts, as I've seen people on this site genuinely believing what's being claimed.
The only reason I'm responding to this, is because this is probably going to be my last post on this site (so if the original poster, or David Cole wants to argue with me, feel free, I won't be here).
I don't really get any benefit out of posting, so it is starting to feel like I'm working for the site without getting paid, so it is time for me to go.
Hope this helped,