Euclid's theorem

There are infinitely many prime numbers. Indeed, for any finite list of prime numbers $p_1,\dots ,p_k$, the integer $N=p_1\cdots p_k+1>1$ has a prime factor, by the Fundamental theorem of arithmetic. None of the listed prime numbers divides $N$, since each divides $N-1$ and a common divisor would divide $1$. Thus no finite list contains all prime numbers. The constructed $N$ need not itself be a prime number.