There are infinitely many primes

Suppose there are only finitely many primes $p_1,\dots,p_k$. Let
$$N=p_1p_2\cdots p_k+1>1.$$

Then $N$ has a prime divisor $p$. This prime must be one of the $p_i$, by assumption. But $p_i$ divides $p_1\cdots p_k$ and divides $N$, so it divides their difference $N-p_1\cdots p_k=1$, which is impossible.

This contradiction shows that the list of primes cannot be finite.