Atheism refers to the absence of a belief in any deities. So it doesn't make any sense to speak of proving atheism in that sense.

But you might be wanting to ask what evidence is there that there is no God. That depends on your understanding of what is meant by the claim that there is no God. For example, suppose we take it to mean that there does not exist any intelligence that created the universe. That proposition is similar to the proposition that the Flying Spaghetti Monster does not exist or that there does not exist a teapot orbiting the Sun. You can't really produce any specific item of evidence in favour of the claim that these things do not exist, but there is absolutely no evidence at all that these things do exist and it's fairly a priori unlikely to think that they exist. So for that reason, lacking a belief that those things exist is eminently reasonable.

On the other hand, you might take the claim that there is no God to mean that there does not exist any omnipotent, omniscient, omnipresent spirit who created the universe and who cares about the well-being of the sentient beings on this planet. It is of course completely obvious that no such being exists, because the existence of such a being is not consistent with the enormous quantity of excruciating suffering that we can observe to have been endured by so many billions of sentient beings on this planet throughout so many hundreds of millions of years of evolutionary history. Perhaps there exists a deity who created the universe and cares about the well-being of the sentient beings on this planet but who is not completely omnipotent. Or perhaps there does exist an omnipotent, omniscient, omnipresent spirit who created the universe but who is totally indifferent to the well-being of the sentient beings on this planet. But you can certainly rule out the possibility that there could exist an omnipotent, omniscient, omnipresent spirit who created the universe and cares about the well-being of the sentient beings on this planet, so for that reason the teachings of the world's major monotheistic religions cannot be completely true.

You can also rule out the possibility that certain forms of classical theism could be true on the grounds that they are incoherent. For example, many classical theists claim that God exists outside time and is completely unchanging, but also that he is a being with a mind with whom you can have some kind of meaningful personal relationship. That's obviously incoherent, so we can safely rule out the possibility that that version of classical theism could be the truth.

There might exist some kind of god somewhat similar to the being claimed to exist by the world's major monotheistic religions. Hardly any atheist would claim to know with absolute certainty that such a god does not exist. But there is absolutely no reason to think that a being of that kind exists, just as there is no reason to think that the Flying Spaghetti Monster exists or that Invisible Pink Unicorns exist. So it is entirely reasonable to simply dismiss any claim that a being of that kind exists.

-Rupert McCallum