How does proof by induction work?

Proof by induction is one of the most important mathematical methods. We take a statement we think is true for all integers n and prove that we're right:

We show something works in a trival, easy case (the base case). Often this is for n=1 or n=0.

Then we make our Induction hypothesis where we simply state that we're going assume our statement is true for some value n = k.

Then we show, often algebraically, that, given our hypothesis, our statement is true for n = k+1. This is the Induction step.

So then the statement is true for all integers bigger than our base case n. We've shown for example, that the base case n = 1 is true so n+1 = 2 must be true due to our induction step. And n = 3, 4, 5... and so for every single integer number.

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