There’s a difference between ‘proving’ something – technically this can only be done in mathematics and, arguably, the law – and ‘demonstrating’ it. Atheists can’t ‘prove’ God doesn’t exist, any more than believers can prove he does. But it is relatively easy to demonstrate how unlikely it is that there’s a God. We can apply the scientific method. Science doesn’t ‘prove’ either – it demonstrates the likelihood of something being the case by looking at the evidence and determining from it whether a phenomenon is probable or improbable.
This is in fact what Christians (and Muslims and Hindus and Jews) do when they decide whether the tooth fairy, Santa Claus and Zeus exist; from the evidence they conclude, quite rightly, that it’s highly improbable.
What believers don’t do, though atheists do, is apply the same test to their own god. They say, ‘well of course these other supernatural beings don’t exist. There’s no evidence for them whatsoever. But as for my God, well, I’m not going to apply the same rigour. I know he exists because, erm, I believe in him, I have faith.’
But faith and belief are not evidence. Warm fuzzy feelings are not evidence of God. Books written by ancient tribesmen and other superstitious people aren’t either. Even the universe itself is not evidence, when its existence can be explained without recourse to him. Similarly the development of life on Earth and human beings themselves; all are better explained by other means, none of which require God.
It is fairly safe to conclude as a result that the reason they don’t require him is because he wasn’t involved. And he wasn’t involved because he doesn’t exist, in just the same way Santa Claus wasn’t involved when you received your Christmas presents this week. This may not be the ‘proof’ Christians and others would like, but it does demonstrate, more than adequately, the improbability of God’s existence.