In a video I watched last night on nuking mathematical mosquitos, Matt Parker gave the following proof of the infinitude of primes: suppose there are finitely many primes. The Green-Tao theorem says there are arbitrarily long arithmetic progressions in the primes, hence there cannot be finitely many primes. Contradiction.
Leaving aside the slightly dubious and unnecessary use of proof by contradiction, it made me wonder whether or not this proof was circular (and Parker himself remarks: "Green and Tao took it as a given that there are infinitely many prime numbers and my pithy proof may very well be circular!"). Namely, is there some fact about the infinitude of primes that is used deep in the proof of the Green-Tao theorem? For instance, in some density arguments or similar?