In Thomas Aquinas' small writing, entitled, "De Ente et Essentia" (or, "On Being and Essence"), the great medieval philosopher sets out to demonstrate that God exists with just one basic metaphysical proof. The argument goes as follows:
1. Changing things exist.
2. Everything that changes is composed of act and potency.
3. No potency can actualize itself.
4. Only Pure Act can actualize being.
5. Therefore, Pure Act exists.
(1) ought to be considered self-evident. It is manifest throughout our experiences that things change. (2) should also be easily accepted once the terms are correctly understood. "Act" (or, "actuality") is what something is, and "potency" (or, "potentiality") is what it could be. For example, a seed is merely a seed in act, but it is a plant in potency.
(3) states that in order for something to change from a state of potency to actuality that there be something that changes it. A seed is merely a seed, but if it has soil, water, and sunlight, it can grow into a plant. One might be tempted to think of quantum fluctuations as an exception to the rule, but even quantum fluctuations cannot arise apart from the energy contained within the quantum vacuum. Hence, there is no exception to be made.
(4) denies the possibility of an infinite regress of changing things. I outlined two of the reasons in my post on the "argument from dependency." For now, just think of it this way: if no potency can actualize itself, then without Pure Act, then nothing would ever be actualized. Even if there were infinitely-many potent things, without Pure Act none of them would be actualized; we would have nothing but infinitely-many non-actualized potencies.
Moreover, at any finite period of time something is changing. Yet, it would take infinite time for an infinite series of things to change anything! On this basis, we can conclude with certainty that the series of changing things is finite, and is grounded in Pure Act.
Now that we have established that Pure Act exists, why should we believe this is a proof of God's existence? In answer to this, we might note that God is defined as a being that is Pure Act, immutable, eternal, one, omnipotent, and omniscient. Let's examine each of the remaining attributes.
Pure Act is immutable. If Pure Act could change, then it would be composed of potency. However, there is no potency in Pure Act. Therefore, Pure Act is immutable (or, unchanging).
Pure Act is eternal. If Pure Act were to come into existence, or go out of existence, then that would require some change in its essence. But as demonstrated above, Pure Act is immutable. Hence, Pure Act is eternal.
Pure Act is one. If there were more than one Pure Act, then there would be distinctions between them. But, distinctions entail limitations, and limitations entail a composition of potency. Again, we have demonstrated that Pure Act is not composed of potency, so it must be one.
Pure Act is omnipotent and omniscient. Partly actual beings, like ourselves, have some power and some knowledge. Something purely actual (re: Pure Act) would therefore possess all power and all knowledge. One might object that Pure Act would also have to possess all ignorance, but this misconstrues the nature of ignorance. Ignorance is simply a lack of knowledge; it's a privation, rather than something positively actual.
With all of the above taken into consideration, we have shown that there exists a purely actual, immutable, eternal, one, omnipotent, and omniscient being. This matches our definition of God, and since this being exists, it can be concluded that God exists.