The first unjustified passage is
"Since it is repeatedly dividing a even number by two, if the series converges, it will sure reach 1. Therefore it is enough to demonstrate that the series converge"
Here "converges" is of course to be taken in a modifed sense of "cycle" (because even if the Collatz conjecture is true, the sequence obviously does not converge in the usual sense). But then it hasn't even been shown yet that $1-4-2$ is the only possible cycle.
"For any starting number N, the series will converge if z is ON AVERAGE greater than the number given by the equation above. In this way the sequence of operations will subtract more than add to the series."
This is not justified either. For it to have any chance of being true, "on average" would have to be quantified way more than just this way. What if, every time it's not greater, then it just becomes way bigger and you can't control it that way ?
I won't go further, because it seems to be again just an endless flow of unjustified claims.
Let me take this opportunity to make a statement about proofs in maths : the point of proofs is that it's not the reader's job to try to correct the proof; the author of the proof has to justify every single one of their claims, if they can't and the reader has an objection to which the "prover" can't answer, then it's not a valid proof.
Of course, the amount to which one has to justify the claims also depends on the public you're aiming at : professional mathematicians are used to certain types of reasoning and so can fill some gaps by themselves when reading paper (for a stupid basic example, if you're proving something to a mathematicall educated audience, you don't have to say "and then by modus ponens we have ..."); but it is still the case that in the prover vs reader "game", the reader is always right, and when a part of a proof is not clear to them, they are right and the prover has to justify that part of the proof.