That argument turns out not to be enough! A counterexample is the set of rational numbers, which has the same cardinality as the natural numbers, even though th

That argument turns out not to be enough! A counterexample is the set of rational numbers, which has the same cardinality as the natural numbers, even though the naturals are a proper subset of the rationals. Some infinite sets can be put into one-to-one correspondence with some of their proper subsets!