St. Thomas Aquinas is famous for what are known as his "five ways," or arguments, for God's existence. In the first three ways, Thomas is specifically concerned with cosmological arguments, a class or family of proofs intended to demonstrate the existence of a First Cause or Necessary Being. Today the cosmological argument is often associated with the so-called "kalam" cosmological argument (kalam, or kalaam, comes from the Arabic word for "reason" or "speech"). This latter argument requires that the universe had a beginning in the finite past. What distinguishes the kalam argument from Thomas's cosmological proofs is that Thomas, following the ancient Greek philosopher Aristotle, leaves the finitude of the universe's past as an open question. Instead, Thomas is interested in establishing the existence of a First Cause in the order of sustaining causes, as opposed to the order of originating causes. After all, it's one thing to ask why something begins to exist, and quite another to ask why it continues to exist. The originating cause of an oak tree might be someone who planted an acorn, for example, whereas the oak tree has sustaining causes (water, soil, sunlight, etc.) that, if removed, would result in the oak tree's destruction.
Below I will summarize the third of Thomas's cosmological arguments. You will notice that whether the universe's past is finite or infinite is irrelevant to the demonstration.
1. Something presently exists. (Premise)
2. Something cannot come from nothing. (Premise)
3. Either everything that exists is contingent, or else there exists at least one necessary entity N. (Premise)
4. Necessarily, there was never a past time at which nothing existed. (From 1 and 2)
5. Possibly, there was a past time at which nothing contingent existed. (Premise)
6. Therefore, a necessary entity N exists. (From 4 and 5)
Reductio ad absurdum:
7. N does not exist. (Assumption)
8. Possibly, there was a past time at which nothing existed. (From 3 and 5)
9. (8) contradicts (4). (Premise)
10. Therefore, (7) is false and N exists. (From 7 – 9)
Premise (1) is obviously true. As Descartes famously put it, "I think, therefore I am." In order to doubt that I exist, I would first have to exist in order to doubt it! Premise (2) is based upon the metaphysical principle that being cannot arise from non-being. Some have attempted to cast doubt on this premise by pointing to quantum physics, in which fluctuations arise spontaneously. However, this is a misleading objection, since the fluctuations only arise from the energy contained within the quantum vacuum. As Christopher Ray points out, "it is a mistake to think of any physical vacuum as some absolutely empty void." Since energy is something, it follows that quantum fluctuations do not constitute a violation of premise (2).
Premise (3) simply distinguishes between two types of entities. A thing can either exist contingently, in which case it is generable and destructible, or else necessarily, meaning that it exists eternally and is indestructible.
Given the truth of (1) and (2), it follows that there was never a past time at which nothing existed, in confirmation of (4). After all, if there were a past time at which literally nothing existed, then nothing could ever come into existence. That would mean nothing would exist right now, but that of course is false.
Premise (5), however, is especially key to the argument, even though it makes a rather benign statement. Even if some contingent thing or other has always existed, there was still the possibility that all contingent things would concurrently cease to exist. Here's an analogy. If every part of a mountain possibly ceases to exist, then the mountain as a whole possibly ceases to exist. Likewise, if every contingent thing possibly ceases to exist (which is true by definition), then it is possible that the sum total of contingent things possibly ceases to exist.
Now, (6) - which states that a necessary entity N exists - follows from (4) and (5), but it isn't immediately obvious why. That's the reason for the premises beginning with (7). We use something called a reductio ad absurdum ("reduction to the absurd"), which refers to an argument that demonstrates the truth of something by showing that its negation is impossible.
(7) merely assumes for the sake of argument that N does not exist. What this leads to is (8): Possibly, there was a past time at which nothing existed. (8) is true because if N does not exist, and it's possible that nothing contingent exists, then we are left with a possible past time at which nothing at all exists, since contingency and necessity are the only two modes of existence. (The only remaining alternative is for something to be impossible, in which case it cannot exist anyway.) The problem, as (9) points out, is that (8) contradicts the conclusion we arrived at in (4), namely, that there was never a past time at which nothing existed. Thus, (10) concludes that the assumption made in (7) is false. Therefore, a necessary (eternal and indestructible) entity N exists.
 For a similar version of the argument, see: Robert E. Maydole, "The Modal Third Way," International Journal for Philosophy of Religion 47, Kluwer Academic Publishers, 2000, pp. 1-28.
 Christopher Ray, Time, Space and Philosophy, Routledge Publishing, 1991, p. 205.