By contradiction, suppose there is a finite amount of natural numbers.
Take the greatest natural number and call it $ n $.
Now consider $ n + 1 $. There are two facts about $ n + 1 $:
- $ n + 1 $ is greater than $ n $
- Since $ n \in \N $, then $ n + 1 \in \N $
This is in contradiction with the statement earlier that there exists a largest natural number.
Hereby it is proven that there are infinitely many natural numbers.