I am not guessing why the two users rejected your edit. I just point out one problem that I see: you changed
$$\forall \epsilon>0 \ \exists n_0 \ \forall n\geq n_0 |a_n-L|>\epsilon,$$
to the "correct" one:
"$\forall L, \exists \epsilon > 0 : \forall N\in\mathbb N, \exists n\ge N : |(x_n)-L| \geq \epsilon,$
which is the negation of the definition of a convergent sequence
$\exists L,\forall \epsilon>0: \ \exists N\in\mathbb N, \ \forall n\ge N: |(x_n)-L|<\epsilon.$"
Although this is mathematically correct, it could happen that the questioner is stuck precisely because they use the incorrect definition. The better action would be to point out in a comment that the definition is wrong.
That being said, that post is really bad. The OP did not give any attempt, and more importantly, the answerers are not even trying to answer using the $\epsilon-N$ definition. I suggest to close and delete it.
Update: Checking the edit history, the question-er actually got the negation almost correct (see edit 1), but it was wrongly edited in 2019 to the current wrong one (edit 3) (Actually, the editor tried to undo their mistake here, but they seem to do it wrong and the suggested edit was rejected). If you made that clear in the edit comment that you are undoing the wrong edit in 2019 and I was the reviewer, I would approve the edit.