Archimedean Property

This page has been proposed for an upcoming wiki collaboration. It is currently under construction, and you can help by adding examples of what you think should be on this page

Let $( S, \circ$ be a closed algebraic structure with a Norm (e.g. real numbers with absolute value as a norm). Then, the norm $n$ satisfies the Archimedean property on $S$ if and only if

$$\forall a, b \in S, n(a) < n(b) \Rightarrow \exists m \in N \text{ such that } n ( m \cdot a) > n (b)$$

Corollary: For all real numbers $r$, there exists an integer $n$ such that $n > r$.

Note: The field of rational functions of $x$ does not satisfy the Archimedean property.

Frequently, Dedicand cuts are used axiomatically to construct the Reals. The Reals definitely have Dedicand completeness. :) The rationals do not.

A la wikipedia:

Real number system has the property that every non-empty subset of R which is bounded above has a least upper bound. This property is called Least Upper Bound Property.

Cauchy completeness is the statement that every Cauchy sequence of real numbers converges. Cauchy Sequence is a sequence whose elements become arbitrarily close to each other as the sequence progresses. Mathematically, for a given $\epsilon>0$, there exists a natural number $N$ such that $|x_m-x_n|<\epsilon\ \forall\ m,n>N$.

For each $n$, let $I_n = [a_n, b_n]$ be a (non-empty) bounded interval of real numbers such that $$I_1 \supset\ I_2 \supset I_3 \supset \cdots \supset I_n \supset I_{n+1} \supset \cdots$$ and $\displaystyle\lim_{n\rightarrow 0} (b_n-a_n)=0$. Then $\displaystyle\bigcap_{n=1}^{\infty} I_n$ contains only one point.

The nested interval theorem states that the intersection of all of the intervals $I_n$ is nonempty.

Source: Springer's "Real Analysis and Applications"

The monotone convergence theorem states that every nondecreasing, bounded sequence of real numbers converges.

The Bolzano–Weierstrass theorem states that every bounded sequence of real numbers has a convergent subsequence.

Prove that Real numbers follow Archimedean Property

Source: Springer's "Real Analysis and Applications"

Prove that Hyperreal Numbers do not follow Archimedian Property

Existence of rational/irrational number between two real numbers

The uncountability of the reals.

Arithmetic Properties

Properties of Inequalities in the Reals

Let $x\in R$. Show that there exists an integer $m$ such that $m\le x < m+1$ and an integer $l$ such that $x<l\le x+1$.

Suppose that $\alpha,\beta$ are two real numbers satisfying $\alpha<\beta$. Show that there exist $n_1,n_2 \in N$ such that $\alpha<\alpha+\frac{1}{n_1}<\beta$ and $\alpha<\beta-\frac{1}{n_2}<\beta$

....

pages 44 to 48