I was teaching a nine-year-old friend about prime numbers. When I asked him if he thought there were finitely or infinitely many primes, he answered confidently that there must be an infinite number. "How do you know?" I asked. "Because I can keep thinking up larger and larger primes. It's easy!" By way of proof, he came up with a new larger prime.
I call this "Naive Induction" (there might be a better term). I am looking for not-too-complicated counterexamples where
It appears to be the case that there are infinitely many members of a set, or (equivalently) that some property is true for all integers, but
It can be demonstrated that there is a largest member of the set, or a largest number with some property.
Any suggestions? Thanks.