Are there infinity twinprimes?

According to my information the evidence, that there are infinity primes is reasoned through the Euclid-Numbers which always are prime: \(P_{1} \times P_{2} \times ... \times P_{n} = N⇒[N+ 1] ∈ Primes\). Since you always have the opportunity, to multiply all the previous primes and add 1, there must be infinity primes.

Now I've heard 2 interesting facts:

1. Euclid-Numbers aren't always prime and
2. It hasn't been proven yet, that there are infinity twinprimes.

It makes good sense, that not all Euclid-Numbers necessarily have to be prime. It's proven, that the Euclid-Numbers aren't divisible by any number between $P_{1}$ and $P_{n}$ (because there will always be a remainder of 1), but what about $P_{n+1}$? There still are other primes, that could be factors of $N+1$. The first Euclid-Number, that isn't a prime is 30031 = 59 × 509 (it's the 6th Euclid-Number). Here the trick doesn't work anymore, because $N+1$ took primes between $P_{n}$ and $N$. So why is this still a prove? Of course it seems very unlikely, that there is an end for the primes, but is it still a prove?

The reason, why this Euclid-Numbers-Trick works is, because the formula makes sure, that the most often used (least) primes can't be a factor of $N+1$ and this makes the probability to get a composite number much worse. But actually we can do the same trick with $-1$ instead of $+1$. if we do $N - 1$, there will be a remainder of $-1$ when we divide it by any integer between $P_{1}$ and $P_{n}$. So $P_{1} \times P_{2} \times ... \times P_{n} = N ⇒ [(N + 1), (N - 1)]∈Primes$ But why isn't this a prove, that there are infinity twinprimes, though it uses the principle of the proof, that there are infinity primes (or am I a professor now 🙂)?