 Solution

If the complete list of prime numbers were $p_1,\dots ,p_k$, then $N=p_1\cdots p_k+1>1$ would have a prime divisor by the fundamental theorem of arithmetic. No $p_i$ divides it, since $N\equiv 1(\mathrm{mod}{p}_i)$. This contradicts completeness of the list. Hence there are infinitely many prime numbers.