From IN Herstein's Topics in Algebra, there is a question (Exercise 1.2.13), a set S is called infinite if there exists a proper subset, A, of that set such that there is a one-to-one correspondence between S and A.
Question: We need to prove that if S has a subset A, which is infinite, then S is infinite.
I am very confused as to how we can prove this but I have tried my hand at proof by contradiction. Note how the question itself does not say if A is a proper subset or not. Also the concept of cardinality has not been introduced yet.
Contradiction: Let S not be infinite. This means there does not exist a proper subset of S, with a one-to-one correspondence between it and S. However we know that A is an infinite set. Which means A has a proper subset, say B, between which there is a one-to-one correspondence.
How do I go further than this?
Best Answer
I would proceed directly. Here's a picture hint of how you could construct a one-to-one correspondence between $S$ and a proper subset.