We all use mathematical induction to prove results, but is there a proof of mathematical induction itself? I just realized something interesting. At schools and universities you get taught mathematical induction. Usually you jump right into using it to prove something like 
$$1+2+3+\cdots+n = \frac{n(n+1)}{2}$$
However.
I just realized that at no point is mathematical induction proved itself? What's the mathematical induction's proof? Is mathematical induction proved using mathematical induction itself? (That would be mind blowing.)

Suppose that $ 1 \in S \text{ and that } n \in S  \Rightarrow n+1 \in S$, and suppose that there is a smallest number $N$ such that $N+1\notin S$...

Supposing that any subset of natural numbers has a minimal element, the above proves the principle of induction in set theory, whether some people like it or not.
