I see this as a false dichotomy. You can have faith and proof and God doesn’t expect us to kiss our brains goodbye or to commit intellectual suicide or to have blind faith (without evidence). If you put God to the test and demand your kind of proof before you’ll believe, He will not accommodate you. God needs not to prove Himself to anyone and has given all the evidence necessary in the Bible. Note: evidence isn't always definitive, compelling, objective, or authoritative.
How do you define faith and proof? Faith can be seen as trusting in what you have good reason to believe and knowledge in action; what’s wrong with that? Everyone has faith in something; faith in science is still faith. You can have faith in yourself and your own reasoning. All knowledge is contingent and begins in faith with some presupposition that cannot be proved, even in math. Proof and evidence are often used interchangeably. Proof can be seen as just an argument or reason to believe: evidence. The apostle Luke said that there “were many infallible [or convincing] proofs” for the resurrection and he was a doctor and must have known something about proof.
There are many philosophical arguments for the existence of God, for example, but you cannot either prove or disprove God beyond a shadow of a doubt; there's no “smoking gun” evidence either way and both theists and atheists are persons of faith. No amount of evidence would make you believe without any possible doubt either way. There is never enough proof or evidence for someone who doesn't want to believe something, even in religious matters.
Note: there are different categories of evidence such as in law, science, philosophy, history, and literature. In science, something isn't held as true unless it is demonstrable and observable (repeatable and measurable); these types of evidence don’t hold water for metaphysical matters and go beyond the parameters of the scientific method. In a court of law, truth (it must be beyond a reasonable doubt, not all doubt), is by oral and written testimony and considered evidence. In history, for example, corroborating texts and artifacts are considered evidence. No one can not believe in God, for instance, due to the lack of evidence of seeing all the evidence in nature and God says no one has an excuse and that this is plain to see.