Prove there is infinite prime
$\mbox{There is infinite prime}$ Assume there is finite number of primes from $p_1 \dots p_k$
and let $n = p_1 p_2 \dots p_k + 1$
$\therefore n - p_1 p_2 \dots p_k = 1$
$P$ is not a prime
let $q \mid P$ but $q \nmid p_1 p_2 \dots p_k$ otherwise $\frac{P}{q} - \frac{p_1 p_2 \dots p_k}{q} = \frac{1}{q}$ which is impossible
since left side is integer and right side is non integer
Therefore $P$ must be a prime and has not prime factor from $p_1 p_2 \dots p_k$
Hence, there are infinite primes