Infinitely many prime numbers

Euclid's proof

By contradition, let's consider that the set $\{ p_n \}$ of prime numbers has finitely many primes, with some cardinality $N$. Therefore, there is a highest prime $p_N$. Now take the new number

\begin{equation} p = \left(\prod_{i=1}^N p_i \right) + 1 \end{equation}

If there are only finitely many primes, any number should be able to get expressed by

\begin{equation} n = \prod_{i = 1}^N (p_i)^{c(i)} \end{equation}