Concrete Mathematics Chapter 1 Warmups

Exercises 1: All horses are the same color; we can prove this by induction on the number of horses in a given set. Here's how: "If there's just one horse then it's the same color as itself, so the basis is trivial. For then induction step, assume that there are n horses numbered 1 to n. By the inductions hypothesis, horses 1 through n - 1 are the same color, and similarly horses 2 through n are the same color. But the middle horses, 2 through n - 1, can't change color when they're in different groups; these are horses, not chameleons. So horses 1 and n must be the same color as well, by transitivity. Thus all n horses are the same color; QED." Whant, if anything, is wrong with this reasoning? \(\sum_{1} = x_i y_i\)