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.)