Counterexamples to "Naive Induction" 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.

This isn't what you're looking for, but it's semi - related.  
Euler's polynomial $n^2 + n +41$ generates primes for integers $n = 0$ to 39.  It seems when you start plugging in numbers and checking that this always gives a prime!
