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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.