Explain what is wrong with the following proof by mathematical induction that all horses are the same colour: Clearly all horses in any set of 1 horse are all the same colour. This completes the basic step. Now assume that all horses in any set of $n$ horses are the same colour. Consider the set of $n+1$ horses, labeled with the integers $1,2, \ldots, n+1$. By the inductive hypothesis, horses $1,2, \ldots, n$ are all the same colour, as are horses $2,3, \ldots, n+1$. Because these two sets of horses have common members, namely, horses $2,3,4, \ldots, n$, all $n+1$ horses must be the same colour. This completes the inductive argument.