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.
Contents
- Forms of Completeness
- Least Upper Bound Property
- Cauchy Completeness
- Nested Intervals Theorem
- Monotone Convergence Theorem
- Bolzano Weierstrass Theorem
- Archimedean Property
- Example 2
- Example 3
- Example 4
- Example 5
- Example 6
- Example 7
- Example 8
- Example 9
- Example 10
- Example 11
- Example 12
- Example 13
- Example 14
- Example 15
- Example 16
- Example 17
- Example 18
- Example 19
- Example 20
Forms of Completeness
Frequently, Dedicand cuts are used axiomatically to construct the Reals. The Reals definitely have Dedicand completeness. :) The rationals do not.
A la wikipedia:
Least Upper Bound Property
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
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 \(|x_m-x_n|<\epsilon\ \forall\ m,n>N\).
Nested Intervals Theorem
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"
Monotone Convergence Theorem
The monotone convergence theorem states that every nondecreasing, bounded sequence of real numbers converges.
Bolzano Weierstrass Theorem
The Bolzano–Weierstrass theorem states that every bounded sequence of real numbers has a convergent subsequence.
Archimedean Property
Prove that Real numbers follow Archimedean Property
Source: Springer's "Real Analysis and Applications"
Example 2
Prove that Hyperreal Numbers do not follow Archimedian Property
Example 3
Existence of rational/irrational number between two real numbers
Example 4
The uncountability of the reals.
Example 5
Arithmetic Properties
Example 6
Properties of Inequalities in the Reals
Example 7
Let \(x\in R\). Show that there exists an integer \(m\) such that \(m\le x < m+1\) and an integer \(l\) such that \(x<l\le x+1\).
Example 8
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 \(\alpha<\beta-\frac{1}{n_2}<\beta\)
Example 9
....
Example 10
Example 11
Example 12
Example 13
Example 14
Example 15
Example 16
Example 17
Example 18
Example 19
Example 20