What definition of finite are you using?

Is it that a set is finite iff there is a bijection defined on it to a set of consecutive naturals with a largest element.

If so let be a bijection from to then is a bijection from to . Now we just need to introduce a renumbering function on so these are consecutive integers.

Consider:

Where is the indicator or charateristic function of (that is it is for any element of and zero otherwise).

Then :

is a bijection of to a set of consecutive integers with a largest element(you will have to fill in the detailed justification of this).

CB