How do we prove that the Church is infallible without falling into circular reasoning?

Look at how John opens the Apocalypse. First the Father, then Jesus Christ, then the church, then John himself. That is his line of authority. He is an old man, exiled to Patmos, in his eighties or nineties in 95 AD, writing fantastic symbolic things nobody had ever seen, frogs coming out of men's mouths, a woman riding a beast. Why should anyone believe him? He is alone on an island with no standing among the leaders of the world. So he appeals as high as he possibly can, to the Father: every word I write here comes from the Father, through Jesus, through the church, to me. If you do not want to believe it, that is your choice, but that is my chain of authority. On top of that, I walked with Peter, Andrew, and James; read the book of Acts, we were doing miracles almost every day, which is how people knew we were from God. Yes, some of it is circular, but if it is true it can be circular; being circular does not make it false. It is the world that demands proof of everything. The Pharisees watched Jesus cure the sick and raise the dead and still wanted some grand celestial sign, because their hearts were wrong, and he knew that even if he turned the moon to green cheese they would not believe. For those ready to take it by faith, the miracles are enough. The only full proof you will have is when you see Jesus at the second coming.