> (seems hand wavey, and probably wrong though I can't figure out where the flaw is)
The flaw is here:
> Since there are infinite sequences of composite numbers, and each composite number is the product of at least two prime numbers (for example, 12 is the product of 2 and 6, which are both prime numbers), there must be an infinite number of prime numbers as well.
This doesn't establish that the prime numbers are distinct. The infinite sequence of even numbers has a single prime number in it, for example.
The flaw is here:
> Since there are infinite sequences of composite numbers, and each composite number is the product of at least two prime numbers (for example, 12 is the product of 2 and 6, which are both prime numbers), there must be an infinite number of prime numbers as well.
This doesn't establish that the prime numbers are distinct. The infinite sequence of even numbers has a single prime number in it, for example.