Euclid's Theorem

Theorem
There are infinitely many prime numbers.

Proof by Contradiction
Assume that there are only finitely many prime numbers, denoted $$ p_1,p_2,\dots,p_n$$, where $$p_n$$ is the largest prime.

Consider the number: $$\left( \prod_{i=1}^{n} p_i \right) +1$$. This number must be composite, since it is clearly larger than the largest prime. So, we must have at least one prime divisor from our list $$ p_1,p_2,\dots,p_n$$. Denote this prime $$p_j \,\!$$. But, since $$ p_j|\prod_{i=1}^{n} p_i $$, we have that $$p_j|1 \Rightarrow p_j \leq 1 \,\!$$, a contradiction.

Thus, there are infinitely many prime numbers.

Q.E.D.