Euclid's Theorem

Theorem
There are infinitely many prime numbers.

Proof by contradiction
Assume, by way of contradiction, that there are only finitely many prime numbers, denoted $$ p_1,p_2,\dots,p_n$$, where $$ p_n $$ is the largest prime. Now, consider the number given by $$\left( \prod_{i=1}^{n} p_i\right) +1$$. This number must be composite, since it is clearly larger than the largest prime. So, we must have a prime divisor from our list $$ p_1,p_2,\dots,p_n$$. But, since $$ p_j|\prod_{i=1}^{n} p_i$$ for any $$j \in \{1,2,3,\dots,n\}$$, we have that $$p_j|1$$, which is clearly a contradiction.

Thus, there are infinitely many prime numbers.

QED