Euclidean prime number theorem

The proof of the Euclidean theorem is simple. Suppose there exist only finitely many prime numbers $p_1,\ldots,p_k$. Consider the number $N=p_1\ldots p_k+1$. Since $N>1$ it must be divisible by a prime number $p$, which equals some $p_i$ due to the finiteness of the amount of prime numbers. Hence $p=p_i$ divides $N=p_1\ldots p_i\ldots p_k+1$, and thus $p_i$ divides 1. This contradiction shows that there must be infinitely many prime numbers.