About Euclid's proof of infinite primes..... I was checking that if product of first n primes+1 gives a prime again is true to how many n
For example $$2+1=3$$ is a prime$$2\times 3+1=7$$ is a prime$$2\times 3\times 5+1=31$$ is a prime$$2\times 3\times 5\times 7+1=211 $$ is a prime$$2\times 3\times 5\times 7\times 11+1=2311$$ is a prime $$2\times 3\times 5\times 7\times 11\times 13+1=30031$$ is composite
So prime chain is broken and further steps will give composite no.s only
Now as I understood from proof of infinite primes Euclid said multiply all primes and add 1 and you will get another prime. It proves that there are infinite primes
Then my question is how can we guarantee that the resulting number is prime??
Or it should be that we will get another prime dividing the resulting number
Please help me to clear my confusion!!!!

Euclid's proof of infinite primes is not to give a method of generating infinite primes, it is to prove that there are cannot be a finite set of primes, which are two very different things. 
It is non-constructive and the entire basis of the proof is to show that assuming a finite set of primes will lead to a contradiction by definition since $a \times b \times c \times d \ \times .... + 1$ cannot be divisible by any of the product terms if $a, b, c, ... \in \mathbb{N}$ as this would say that $\frac{1}{a}$ is an integer. 
