A set is infinite iff there is a one-to-one correspondent with one of its proper subsets? Maxwell Rosenlicht claims in "Introduction to analysis" that a set is infinite if and only if it may be placed into one-to-one correspondence with a proper subset of itself.
He says this is self-evident because a finite set cannot be placed into a one-to-one correspondence with a proper subset of itself (because it has fewer elements), and whilst this is reasonable - I cannot follow Rosenlicht in that "the above therefore follows obviously". Why must a set be infinite just because of some property of finite sets?

This is known as
the Dedekind definition
of a set being infinite.
Here is more:
http://en.wikipedia.org/wiki/Dedekind-infinite_set
As an exercise,
you might try to show that
this is equivalent 
to the definition
stating that
the set,
or some subset of it,
can be placed into a
1-1 correspondence with
the positive integers.
