If you're trying to prove that He doesn't exist, you are trying to prove a negative. That's a very difficult thing to accomplish.
We have a wealth of scientific evidence which describes how the universe "just happened"--but that scientific evidence does not prove the non-existence of God; it merely shows how the beginning of the universe could have led to our existence.
But if you're trying to prove that God does exist, what then?
What do you cite as evidence of His existence? Is there any factual evidence of His existence? No; and I think things are that way by design. (Just to clarify, I do believe in God.)
So, let's look at a few things:
1) It is a basic tenet of Christianity that without faith, God is nothing. (It may be a basic tenet of Islam and Judaism as well; I don't know.)
2) Proof denies faith. I don't need to have faith in the existence of the chair I'm sitting on; the fact that it supports my adipose carcass is proof that it does.
3) People who believe in God will believe in Him despite an absence of evidence; people who believe He does not exist will not be convinced otherwise.
Just one example would be the debate over evolution versus creation: a properly scientific examination of the origin of advanced life on Earth shows us that evolution must have taken place; but accepting this theory does not necessarily require that God be removed from the process. There is no reason this is so; God may have used evolution as His tool for creating "all creatures, great and small". From this perspective, evolution, creationism, and intelligent design are not at all incompatible.
But why would He bother? Why not just wave His hand and make it all appear?
...why not, indeed? How can we know that that is not exactly what He did?
By definition, God is omnipotent: He knows all, and can do whatever He wishes with the universe. Nothing is beyond Him. If this is so, then it would not be beyond His capacity to create a universe which looks, to any and all inspection, as if it "just happened"...particularly to the creatures which He created to inhabit it.
God could have created this universe five seconds ago, and none of us would know the difference. For all I know, this sentence is the first one I'm actually typing....
Light that is arriving at Earth right now, from distant galaxies, would have been created "already on the way". Fossil evidence would exist as if the world had actually accreted, cooled, and sprouted life. All our history up until now would have been created from scratch, but we could not prove it.
"How many angels can dance on the head of a pin?"
The problem, as you can see, is that fundamentally it really doesn't matter whether or not God exists. We can't know.
But why? Why should He go to all the trouble of creating us, and this universe--and demand that we worship Him!--and not let us know He is really there?
"Demand that we worship Him"--God figures in three of the world's major religions, and in all of them it's implicit that failure to worship God can lead to eternal damnation of one sort or another. So why all the mystery? If He wants us to worship him, well, we would definitely worship Him if He made His presence known to us!
According to the Christian Bible, Lucifer--Satan--was once an angel; he disagreed with God; as I recall, he felt that humans should be ruled by angels.
Humans have free will; this is something which even angels do not have. We can choose to worship God, or not to, as suits us. Our religions lay out the rules but we are under no compulsion whatsoever to follow them.
Free will is the greatest gift that God could have given us. As far as we know, ours is the only species which has it. The animals do not have free will. I infer that the angels do not have free will. Only God, and humans, have free will.
Accepting that, then, why would God take free will from us by proving His existence to us? If He proved His existence, we would not reasonably be able to choose to worship, or not to worship, as suited us: only an insane person would risk eternal damnation by refusing to worship a god he could visit in the flesh!
(For the sake of discussion I am ignoring those people who are warped or cracked enough to choose to serve Satan. I don't know what the benefits of Satan worship are, but considering that Lucifer hates all humans, I don't see how an afterlife with Satan could be beneficial.)
So: it's impossible to prove God exists. It's impossible to prove He doesn't. Unless (until?) God chooses to reveal Himself to the world, it's all going to be a matter of faith. And that's how it should be.