I was wondering how Euclid showed that there are infinitely many primes by generating a prime number from finitely many primes, and if it could be used to answer if there are infinitely many pairs of primes whose difference is 2. I show my approach in my - short - article here (I know I should copy here the relevant bits, but the article is really short).
My question: why is it not proved like this already? Am I missing something?
Update:
It seems I was too lazy to check for counterexamples. I removed my article. Thank you for the lot of feedbacks.
Euclid proof states that $\psi_n +1$ is itself prime or it contains new primes in its factorization, for example
$$2\cdot 3 \cdot 5\cdot 7 \cdot 11\cdot 13+1=30031=59 \cdot 509$$