Primes

Euclid's proof:

Suppose that \(p_1=2 < p_2 = 3 < ... < p_r\) are all of the primes. Let \(P = p_1\cdotp_2...pr+1\) and let p be a prime dividing P;

then p can not be any of \(p_1, p_2, ..., p_r,\) otherwise p would divide the difference \(P-p_1p_2...p_r=1\), which is impossible.

So this prime p is still another prime, and \(p_1, p_2, ..., pr\) would not be all of the primes.