Nontrivial Definitiom of an Infinite Set???

Question

Can I get an example of an Infinite Set which demonstrates the defintion....

Answer 1

In set theory, an infinite set is a set that is not a finite set. Infinite sets may be countable or uncountable. Some examples are:

the set of all integers, {..., -1, 0, 1, 2, ...}, is a countably infinite set; and
the set of all real numbers is an uncountably infinite set.

The set of natural numbers (whose existence is assured by the axiom of infinity) is infinite. So any set that can be put in one-to-one correspondence with the set of natural numbers is an infinite set.

Ex. Show that there are an infinite number of points on the segment form zero to one, [0, 1]
The set of fractions 1/1, 1/2, 1/3, 1/4, ... etc. all lie on the segment [0, 1] and that set of fractions is infinite because their denominators can be put in one-to-one correspondence with the set of natural numbers.

Answer 2

Set of integers (.....1,0,1,2.....)
Set of real numbers is an uncountable infinite set.
Natural numbers are infinite.