How: by what method?
If I tried to be unbiased I might unknowingly be unsuccessful, so by talking to you guys I get a balanced view of the possible causes of error. Yet all the arguments you use are dealt with in the Bible or elsewhere.
For example, Jeremy says "how do you know Ignatius knew John?" so I google that question and find that although there's no direct evidence that he did, the theme of some of his letters was the same as one of John's letters. See 2 John v.7, which is about those who were not acknowledging Jesus Christ as coming in the flesh (ie they claimed Jesus only seemed to exist). Ignatius deals with this heresy in his epistle to the Trallians:
"Stop your ears, therefore, when any one speaks to you at variance with(18) Jesus Christ, who was descended from David, and was also of Mary; who was truly born, and did eat and drink. He was truly persecuted under Pontius Pilate; He was truly crucified, and [truly] died, in the sight of beings in heaven, and on earth, and under the earth. He was also truly raised from the dead, His Father quickening Him, even as after the same manner His Father will so raise up us who believe in Him by Christ Jesus, apart from whom we do not possess the true life."
More pertinent to Jeremy's question is that Polycarp is mentioned in a writing by his disciple Irenaeus, who says that Polycarp was taught by John and some of the other apostles:
"Polycarp was not only instructed by apostles, and conversed with many who had seen Christ..." (Against Heresies 3:3:4)
Hope that answers yours and Jeremy's questions.