The thing about proving a negative is that you don't have to. As in law, one only needs to prove 'reasonable doubt'. In this matter, there is more doubt than evidence. There is no reliable primary source for the existence of the biblical Jesus as an historical person. None of the documents we have are contemporary - the earliest Gospel (Mark) dates from 66CE at the oldest. There are no Roman records (which, if Yeshu'a had stood trial before the Procurator, there should be). Any references in the Talmud or other Jewish writings are either later musings on the damage Christianity did to Jews, or so vague as to be useless except as subjects of academic debate. Passages in Josephus are obvious later interpolations.

In short, reasonable doubt!