Euclid's proof of infinitely many primes

Contradiction. Suppose finitely many primes $p_1,\ldots,p_n$. Let $N=p_1\cdots p_n+1>1$.

Prime divisor. $N$ has a prime divisor $p$, which is one of the $p_i$.

Contradiction. $p\mid N$ and $p\mid p_1\cdots p_n$, so $p\mid1$, impossible.

Conclusion. There are infinitely many primes.