Prove that every set that has an infinite subset is infinite.

Also, prove that every subset of a finite set is finite.

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- March 31st 2009, 02:05 PMnoles2188Prove: every set with an infinite subset is infinite
Prove that every set that has an infinite subset is infinite.

Also, prove that every subset of a finite set is finite. - March 31st 2009, 02:12 PMPlato
- March 31st 2009, 03:07 PMnoles2188
Here is the definition from the text: "The statement that the set A is infinite means that there is a nonempty proper subset B of A such that there is a one-to-one correspondence between A and B; A is finite means that A is not infinite."

- March 31st 2009, 04:22 PMPlato
Suppose that is an infinite and we want to show that is infinite.

By the definition and there is a bijection such that .

It should be clear that is a proper subset of .

Define a function as

Now your task is to show that is a bijection.