Yup, you came in and said it before I could. Knowing how many counterexamples there could be doesn't help very much if you don't have any bound on their size. pmiller2 seems to assume that there is some way to identify the two candidates, but the problem is that there isn't; the proof is nonconstructive. Nonconstructive proofs are a thing, and it's often important to distinguish whether a given existence proof is constructive or not.