__Archimedean Property__:

** Theorem:** If , then for any there exist such that .

** Proof:** When , the theorem is evident. For . Let the theorem be false, so that . Thus, is a non-empty set bounded above (for ). Therefore, by completeness axiom has the supremum. Let . Then . Thus, is also an upper bound of . But . This contradicts that . Hence, the assumption is false, and so the statement of the theorem is true.

** Corollary 1:** For any such that.

It follows from the theorem, on replacing by and taking for .

** Corollary 2:** The set is bounded below but unbounded above.

** Corollary 3:** For any there exist such that .

** Corollary 4: **For any there exist such that .

** Corollary 5:** For any there exist unique such that .

** Proof: **By completeness axiom the set being bounded above has the suprema, say . Thus, , i.e. , where . Since is a suprema therefore, it is unique.

The above integer is usually denoted by and is called the integral part of the number .

** Corollary 6:** For any there exist unique such that .

** Corollary 7:** For any there exist unique such that .

** Example:** For any there exist unique such that .

**For real number by corollary 5, there exist unique such that**

__Solution__:

Or

Or

i.e.

The above example illustrates that every positive integer can be uniquely expressed as , for unique .

Such unique representation of natural members are sometimes very helpful in investigating the enumerability of certain sets.