Show that there are infinitely many positive prime numbers.

1. Let us assume that there are a finite number of positive prime numbers namely, $p _1, \space p _2, \space p _3 \space ..... \space p _n$, such that $p _1 < p _2 < p _3 \space ..... \space < p _n.$
2. Let $x$ be any number such that,
$x = 1 + \left(p _1 \times p _2 \times p _3 \times ..... \times p _n\right)$
Observe that $\left(p _1 \times p _2 \times p _3 \times ..... \times p _n\right)$ is divisible by each of $p _1, \space p _2, \space p _3 \space ..... \space p _n$ but $x = 1 + \left(p _1 \times p _2 \times p _3 \times ..... \times p _n\right)$ is not divisible by any of $p _1, \space p _2, \space p _3 \space ..... \space p _n$.
3. Since $x$ is not divisible by any of the prime numbers $p _1, \space p _2, \space p _3 \space ..... \space p _n$, therefore, $x$ is either a prime number or has prime divisors other than $p _1, \space p _2, \space p _3 \space ..... \space p _n$.