real number system 11

In the preceding posts, we have seen some properties of real numbers.

(1) A cut of a real number line satisfies the condition of Dedekind cut in which there exists $\max S$  and does not $\min T$ ,  or there exists $\min T$  and does not $\max S$ .

(2) Bounded monotone real number sequences will converge.

(3) If a set of real numbers is bounded above or below, then it has a supremum or a infimum.

(4) If $a$ and $b$ are positive real numbers, then there is an $n\in \mathbb{N}$ such that $na>b$. (It is called the Archimedean property. ) In an addition, nested intervals have a limit real number.

By the strict proofs, we will get that these four properties are equivalent. Therefore, we are able to adopt which properties as an axiom of completeness.

When four properties are apposed, you will feel a similar work, won't you?

This is a last post of 2013. Best wishes for a happy new year  2014 ! !