Prove: Natural numbers do not have the greatest prime number

Prove: Natural numbers do not have the largest prime number

Proof :
Suppose there is the largest prime number N of natural numbers,
then add 1 to the product of all prime numbers less than or equal to N to construct M
M = 2*3*5*7*11*13*...*N+1 

It can be seen from the above that all known prime numbers cannot be decomposed into M, that is, if all known prime numbers are divided by M and the remainder is 1, there are only two possible conclusions.
Conclusion 1: M is a prime number, and obviously M>N
or conclusion 2: There are prime numbers larger than N that can decompose M.

Either conclusion contradicts the assumption that N is the largest prime number.
So proved.