Here's something else I didn't know: According to the NAB, when Jesus heals the leper and tells him, "See that you tell no one, but go show yourself to the priest, and offer the gift that Moses prescribed; that will be proof for them,"
the Greek can also mean "that will be proof against them." It is not clear whether "them" refers to the priests or the people.
Why write clearly when you can write in Greek?
Proof has, I'd say, an uneasy relationship with faith. Proof is a matter of reason, and it leads to direct knowledge; faith is an acceptance of someone else's knowledge. Proof and faith work together in the life of the Christian, but they are contraries. You can't prove something you must accept on faith, and (strictly speaking) you can't believe something you've proven (though you might only believe, rather than know, you've proven it).
In practice, certainly, there have been a lot of disputed proof claims in Christianity. St. Thomas's five "proofs" of God's existence aren't going to be universally accepted or rejected any time soon. The term "proof texting" is often used derisively, even though all Christian traditions quote Scriptural verses to prove things.
Jesus Himself offered various kinds of proofs of Who He Is. He fulfilled the prophecies of the Messiah, a kind of proof for those who have eyes to see. He countered the Pharisees' arguments against Him by quoting Scripture, and He constantly pointed out evidence of His identity to His disciples that they would have otherwise missed.
At bottom, though, Jesus wasn't big on detailed debate and formal demonstration. He Is Who He Is, and He commands His disciples, not to follow His line of argument, but to follow Him.
Jesus is the Truth, believe it or not. His works are proof enough of that, which makes His works proof enough against all who don't believe.