Question

Find the error in the following proof that all horses are the same color.
CLAIM: In any set of h horses, all horses are the same color.
PROOF: By induction on h.
Basis: For h = 1. In any set containing just one horse, all horses clearly are the same color.
Induction step: For k  1 assume that the claim is true for h = k and prove that it is true for h = k + 1 . Take any set H of k + 1 horses. We show that all the horses in the set are the same color. Remove one horse from this set to obtain the set H1 with just k horses. By the
induction hypothesis, all the horses in H1 are the same color. Now replace the removed horse and remove a dierent one to obtain the set H2. By the same argument, all the horses in H2 are the same color. Therefore all the horses in H must be the same color, and the proof is complete.

Answer 1

"...prove that it is true for h = k + 1 . Take any set H of k + 1 horses. We show that all the horses in the set are the same color. Remove one horse from this set to obtain the set H1 with just k horses...."

The induction process requires to prove that given the proposition is true for k, then k+1 is true.
The above prove is proving that given k+1 is true, then k is true. So k decreases, and does not help to prove that k+2 ... is true.