Infinite primes

#Problem 10
Prove that infinitely many prime numbers are possible.

Solution Scheme and approach
Let if possible there are a1,a2...an number of finite primes.
Consider another number
N=(a1*a2*a3*...*an)+1
Now we can say,
N is such a number which can be divide by all primes but always gives a remainder 1. Hence, contradiction.
Because any number>1 should be divisible by at least one prime number.

=>Infinite primes are possible. Q.E.D.