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!!!!

The proof relies on the fact that every prime is in that product, and that a prime can't divide both a number and that number plus one. 
Assume there are finitely many primes.  If $c$ is their product, then $p$ divides $c$ for any prime $p$.  Therefore $p$ does not divide $c+1$ for any prime $p$.  This is a contradiction, every integer has a prime divisor (for primes, this divisor is itself). 
Note that $30031 = 59 \times 509$. Neither $59$ nor $509$ were included in your product and so there is no contradiction. The proof doesn't claim that this product of primes plus one construction always yields a prime. 
