What is the error in the following proof that 2=1 ? Consider the equation a=b. Multiply both sides by a to get a2=ab. Subtract b2 from both sides to get a2b2=abb2. Now factor each side, so (a+b)(ab)= b(ab), and divide each side by (ab) to get a+b=b. Finally, let a and b equal 1, which shows that 2=1. What is the error in this proof by induction? NOTE: Be precise about where the error is. This requires careful thinking. The proof almost works, but... Claim: In any set of h horses, all horses are the same color. The proof is by induction on h. Base Case: h=1. If the set has just one horse, then all horses in the set have the same color. Inductive Step: For k1, assume that the claim is true for h=k. Then prove that it is true for h=k+1. Take any set of k+1 horses. We show that all the horses are the same color within this set. Remove one horse from the set to obtain a subset with just k horses. By our inductive hypothesis, all the horses in this set are the same color. Now replace that horse and remove a different horse. By the same argument, all of the horses remaining are the same color. Therefore, all the horses in the original set must be the same color