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- Feb 9th 2009, 09:43 PMwannabeguruSet theory problem
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- Feb 10th 2009, 01:43 AMclic-clac
Hi

Of course the proof can depend on how you defined "infinite", but, basically, we can always say that__a infinite set minus a finite subset of its elements is still infinite (and by the way non empty)__

Then, using the axiom of choice, you can choose an element in and name it (in your head)* Since , you can choose an element in and name it , again and you can choose... You can repeat a countable amount of times this operation using the axiom of choice and .

Finally, an injection would be which proves what you wanted.

*Why in your head? Well we can rename the elements in but that may not be very funny : if has already elements that are named for a that would be a problem.

One way to do would be to consider and say that is a bijection, so we can use instead of in the proof.