That is Aquinas borrowing from Aristotle. But there are two problems with the argument. First, while a final cause is far more emotionally satisfying than an infinite regress, both are logically on the same footing. Aristotle simply asserted that an infinite regress was untenable, but he never offered an argument for that. Secondly, even if you accept that the argument is valid it does nothing to prove that a Christian or any other God exists; it only proves there was an unknowable first cause for our universe. But that cause could be some mindless mechanism as well as God worthy of worship. With this argument the multiverse could have as easily have been the result of an unknowable primordial creature defecating.
The truth is that no argument from empirical data or sheer logic has ever successfully proven that a God exists, let alone the God of Judaism and Christianity. Because such a God is posited to exist OUTSIDE of our universe and our realm of knowledge, we can never in any sense prove such an existence.
As many clergymen will correctly assert, religion is a matter of FAITH, not KNOWLEDGE. Personally, I BELIEVE that God exists simply because I as an individual find myself unable to believe otherwise. But there is no way I could philosophically or logically argue for my belief.