The question is: " Prove that if S is denumerable, then S is equinumerous with a proper subset of itself" To begin, I am confused with the term denumerable because the text book gives some diagram which is throwing me off. So can somebody clarify this for me: A set S is denumerable if S and N (natural numbers) are equinumerous. That is, their is a BIJECTIVE function f: N---->S The text book says N is INFINITE. So if N is infite, that means that S HAS TO BE INFINTE if S is denumerable? Now I'm reading the books diagram like this: Countable sets are FINITE OR DENUMERABLE Infinite sets are UNCOUNTABLE OR DENUMERABLE So is this true: "if a set S is denumerable, then it HAS TO BE INFINITE?"