Here is the definition from the text: "The statement that the set A is infinite means that there is a nonempty proper subset B of A such that there is a one-to-one correspondence between A and B; A is finite means that A is not infinite."
Suppose that is an infinite and we want to show that is infinite.
By the definition and there is a bijection such that .
It should be clear that is a proper subset of .
Define a function as
Now your task is to show that is a bijection.