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" 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?"

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# Proof: If S is denumerable, then S is equinumerous w/ a proper subset

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