Well, the argument you're running (and refuting) has a long history, and is a bit more complicated than you make it out to be. The classical version, advanced by Thomas Aquinas (and coming from Aristotle) argues the following:
1. There are events.
2. All events have a cause.
3. But an infinite series of events would be absurd.
4. Therefore there must have been a first cause (1-3)
5. But the first cause must be self-caused (2,4)
6. Only an omnipotent being can be its own cause
7. Therefore, the first cause must be God.
It's long been one problem with this argument that it only proves some omnipotent being, rather than the Christian God (though that would only require one or two more assumptions). The other problem is that it's not clear why an infinite chain of events is absurd. The response to the second problem is that even if there is an infinite chain of events, the fact that this chain exists needs to be explained, and the only possible explanation is God, etc.
In the end, however, rational argument will never be able to prove to someone the existence of God. The flaw is not so much in the arguments (while I'm suspicious of the cosmological argument, I'm a fan of the ontological), but just in human nature. There are always reasons both for and against a belief, so we rarely find ourselves forced to believe something. And, just as there are psychological reasons why people can find it more comfortable to believe in God, there are such reasons why people find it more comfortable not to believe in God.
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"Die Deutschen meinen, daß die Kraft sich in Härte und Grausamkeit offenbaren müsse, sie unterwerfen sich dann gerne und mit Bewunderung:[...]. Daß es Kraft giebt in der Milde und Stille, das glauben sie nicht leicht."
"The Germans believe that power must reveal itself in hardness and cruelty and then submit themselves gladly and with admiration[...]. They do not believe readily that there is power in meekness and calm."
-- Friedrich Nietzsche
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