1. Every dependent being derives its existence from some other being.
2. The series of dependent beings either proceeds to infinity, or has a first being.
3. The series of dependent beings cannot proceed to infinity.
4. Therefore, a first being exists.
This argument is really a summary of Thomas' proof as found in De Ente et Essentia. I like this version, since it doesn't rely on "causation" per se, with all of its alleged deterministic connotations.
The first premise has been defended explicitly since at least the time of Parmenides. Ex nihilo nihil fit: out of nothing comes nothing. If there were literally nothing, then not even the potentiality for something to come into being would exist. But, since something exists, it must be either dependent or self-existent.* We certainly observe dependent things in the world. An oak tree is dependent on the acorn, for example.
The second premise doesn't involve a temporal succession of events. Rather, the claim is that there is a series of beings that entail rank or source. Hence, it is considered hierarchical, instead of temporal.
Is it possible that this series is infinite? Several reasons are given against this conclusion. For one, a self-existent first being seems to be required in order for any dependent beings to exist. Just as a house needs a foundation, else the entire structure collapse, dependent beings likewise require something to hold them up. Moreover, if the series of dependent beings is infinite, then an infinite series would be sustaining something within a finite period of time. However, it would take infinite time for an infinite series to do anything. Hence, the series itself must be finite and therefore is grounded in a first being, in confirmation of (3).
I think this is a very reasonable conclusion to make. One might object that in a finite space, there are infinitely-many points. Of course, this argument is undermined by the fact that mathematical points are abstract and don't possess any physical dimensions. Furthermore, a finite space still has definite first and last points; so if there is any analogy between the two, a first being is still required.
The most difficult part of this argument for the Thomist, in my view, is in making the inference from a first being to the claim that this being's existence and essence
are identical. This conclusion does appear to follow from the argument above, though. The reason why is that if the first being is not dependent on anything else, then it cannot derive either its existence or its essence from another. Allow me to put this more explicitly.
The difference between a real unicorn and an imaginary unicorn is that the former is instantiated in actuality. This doesn't fall prey to the Kantian maxim that existence is not a predicate, since we're not merely adding existence to something purely conceptual. Rather, we're considering an a posteriori claim. From this it follows that a unicorn's existence is dependent on something else that brings about its existence. But, we have already seen that the series of dependent beings cannot proceed to infinity. If a being's existence is distinct from its essence, its existence is either brought about by some external being or by its own essential properties. The problem here is that nothing can be brought about by its own essential properties.** The difficulty with the former is that the first being does not derive its existence from anything else; otherwise, it wouldn't be first, which is a self-contradiction.
The conclusion seems to follow, then, that the first being's existence and essence are identical. The reason I say this is a difficulty (although there are certainly solutions) is that the divine attributes that are later inferred about this first being do appear to be distinct. Goodness seems logically distinct from power, knowledge from aseity, and so forth. The Thomist stresses the doctrine of analogy, which views the attributes of God as one and the same, so that the goodness of God really is the same as His power, etc. It's obvious that in us these characteristics are different, but what about in the divine essence?
*Something is self-existent if it exists by a necessity of its own nature. This shouldn't be confused with self-causation.
**This solution has the same problem as with self-causation. Something must already exist in order to bring about its own existence, which is absurd.