Take the largest prime number P_n. We prove there is a prime number greater than any prime P_n

the sheer volume of numbers not exceeding primorial and not divisible by one of the prime numbers. Take any of these numbers, one of the two or is it a prime number is not included in these prime numbers or composite divisible by prime numbers are not included in these prime numbers. But that would be a composite number, you need to have the prime numbers are not included in these. Hence, no matter how large does not have a prime P_n, there is always a prime number greater than P_n.

Proved a prime number greater than any prime P_n. The number of prime numbers is infinite

the sheer volume of numbers not exceeding primorial and not divisible by one of the prime numbers. Take any of these numbers, one of the two or is it a prime number is not included in these prime numbers or composite divisible by prime numbers are not included in these prime numbers. But that would be a composite number, you need to have the prime numbers are not included in these. Hence, no matter how large does not have a prime P_n, there is always a prime number greater than P_n.

Proved a prime number greater than any prime P_n. The number of prime numbers is infinite

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