I was recently explaining to someone how to prove that there are infinitely many prime numbers, and I mentioned to them that it's not immediately obvious, upon first encountering the natural numbers, that there should be infinitely many primes. Their response was along the lines of, "what? Shouldn't it be obvious? If there are infinitely many natural numbers then shouldn't there be infinitely many prime numbers?"
I explained that, just because a set is infinite, that doesn't mean that a subset thereof is infinite, and pointed to the set of natural numbers less than 10 as a counterexample. This was an unsatisfactory counterexample in a way though since the set of natural numbers less than 10 is very obviously not infinite.
Are there some more interesting finite subsets of the natural numbers? Or interesting finite sets of other infinite sets?