What arguments would you use to prove to an atheist that God exists?

I would give the same arguments St. Paul uses in Romans 1: we can tell God exists just by looking at the creation, because we know it could not come into existence by itself, so it had to come from someone with the power to make it, and Paul says God is known to such a degree from creation that he demands honor and thanks from man, so that no one has an excuse. The other reason they know God exists is not just the physical world outside but what is inside them, which Paul gives in Romans 2: they have a guilty conscience when they do wrong. A plant or animal does not seem to have a guilty conscience, but I do, which tells me I am built differently from anything else in creation, so I have two reasons, the creation on the outside and the conscience on the inside, and Paul says again they have no excuse. If those two best reasons are not enough for someone, walk away; do not cater to them, because God already gave the answer and they are the problem, not the reasons. They are looking for someone to cry on their shoulder; give them what they need and let them think about it. That is how you preach the gospel.