Question: What is wrong with the following "proof" by mathematical induction? We will prove that all computers are built by the same manufacturer. In particular, we will prove that in any collection of n computers where n is a positive integer, all the computers are built by the same manufacturer. We first prove P(1), a trivial process, because in any collection consisting of one computer, there is only one manufacturer. Now we assume P(k); that is, in any collection of k computers, all the computers were built by the same manufacturer. To prove P(k + 1), we consider any collection of k + 1 computers. Pull one of these k + 1 computers (call it HAL) out of the collection. By our assumption, the remaining k computers all have the same manufacturer. Let HAL change places with one of these k computers. In the new group of k computers, all have the same manufacturer. Thus, HAL's manufacturer is the same one that produced all the other computers, and all k + 1 computers have the same manufacturer