Induction
Example: All horses are the same color

Let P(n): In a collection of n horses, all horses are the same color.

Prove: For every integer n>=1, P(n). I.e., All horses are the same color.

Proof:

Base case (n = 1): In a collection of only 1 horse, all horses obviously have the same color.

I.H.: Let k>=1 be arbitrary. Assume P(k), i.e., in any group of k horses, all horses are the same color.

Consider an arbitrary group of k+1 horses. Line up the horses.

If we first consider the collection of the first k horses, those k horses are all the same color by the IH.

If we consider the collection of the last k horses, those k horses are all the same color as well.

So then all k+1 horses have the same color.