For those who believe, no proof is necessary.
'For those who believe, no proof is necessary' is self-evident on one level. If somebody already believes something, then he or she will need no additional convincing. That person already believes.
The difficulty there is that the self-evident observation very easily turns into a circular argument: I believe and have faith that unicorns exist. So unicorns definitely exist. So the only justification I need for knowing that unicorns definitely exist is my faith in unicorns.
Faith is just trust and confidence, it isn't a magical way of learning supernatural facts.
Distinguishing truth from fantasy is a major difficulty in this kind of unevidenced belief. Just because somebody has unshakeable faith in the truth of some idea, doesn't make the idea true. Schizophrenics illustrate that every day with how strongly they believe in the truth of their fixed and unshakeable delusions.
The issue that most interests me is the metaphysical one. It isn't whether some particular individual believes in something, whether God, unicorns or ufos. It isn't the importance of the belief and whether it somehow 'works' in that individual's psychology.
The act of publicly making truth-claims about it moves an object of belief into the objective sphere. So the questions concerning the object of belief move from being individual, psychological and subjective, towards being ontological, epistemological and public:
Should other people believe in its literal existence? Are beliefs about it really true?
For those who doubt, no proof is possible.
'Proof' is a very strong word. I'm inclined to agree that no apodeictic (logically necessary) proof of God's existence is possible. Yet philosophical theologians have tried to concoct them for centuries. (Aquinas, most notably.) They haven't had a great deal of success, but the fact that they've made the attempt at all does show that they felt that giving their faith a stronger foundation was important to them. (It didn't 'out' them as atheists, as some people in this thread would suggest.)
'Evidence' or 'persuasive argument' are weaker ideas than 'proof' and attempts to produce evidence of God's existence or persuasive arguments for it might conceivably succeed. Religious experience might arguably constitute such evidence and conceivably the recent cosmological 'fine-tuning' arguments might someday be spun into a plausible cosmological theistic argument.
I agree that many atheists of the more fundamentalist sort probably can never be convinced. They are as fixed against theistic belief as the theistic faithful are locked on its truth. But I believe that many of the more agnostic sort of atheists can be swayed by evidence and argument. But that requires that theists produce the evidence and argument, and not just posture about how superior their theism makes them and how atheists have hurt their feelings by disagreeing.