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 answer might be a bit late but here we go. 
Statement 1: A set is either finite or infinite. (Law of excluded middle)
Statement 2: A finite set cannot be put into one-to-one correspondence with a proper subset of itself. Proof.
Statement 3: So, by statement 1, if there exists a set which can be put into one-to-one correspondence with a proper subset of itself, then it should be an infinite set. 
See this link for more descriptive proof!
