Here is a proof that all horses are the same color Is this a

Here is a proof that all horses are the same color. Is this a true statement? If not, explain the error in the proof. Theorem. All horses are the same color. Let P(n) be the statement In any group of n horses, all of the horses are the same color. Base Case n = 1. In a group of a single horse, clearly all horses are the same color. Inductive step Suppose that in any group of k horses, all horses are the same color. [To show: in any group of k + 1 horses, all horses are the same color.] Consider a group of k + 1 horses: By the inductive hypothesis, the first k horses are all the same color. Also by the inductive hypothesis, the last k horses are all the same color. Thus, all k + 1 horses are the same color.

Solution

Base case is correct but the proof for k+1 horses is incorrect because it fails for k=1 ie k+1=2

Because then last k horses would be just one horse and first k horses would be just one horse so the two horses need not be the same color.

The proof in inductive step works only for k>1