Intereting Posts

How do you orthogonally diagonalize the matrix?
Is there a change of variables that allows one to calculate $\int_0^\pi \frac{1}{4-3\cos^2 x}\, \mathrm dx$ avoiding improperties?
Characterizing all ring homomorphisms $C\to\mathbb{R}$.
Book to prepare for university math?
Halmos, Naive Set Theory, recursion theorem proof: why must he do it that way?
Is $H^2\cap H_0^1$ equipped with the norm $\|f'\|_{L^2}$ complete?
What are the methods of solving linear congruences?
Smallest function whose inverse converges
Fubini theorem for sequences
Fibonacci Generating Function of a Complex Variable
Can a basis for a vector space $V$ can be restricted to a basis for any subspace $W$?
From constrained to unconstrained maximization problem
Plotting a Function of a Complex Variable
Is $\frac{\textrm{d}y}{\textrm{d}x}$ not a ratio?
limit $\lim_{n\to \infty }\sum_{k=1}^{n}\frac{k}{k^2+n^2}$

I’m trying to prove that;

If any Cauchy sequence is convergent in an ordered field F, every nonempty subset of F that has an upperbound has a sup in F.

Let A be a nonempty subset of F that is not a singleton and has an upperbound in F.

Let $a_0 \notin v(A)$ and $b_0 \in v(A)$.

It’s written in my book that for every $e \in P_F$, there exists $N \in \omega$ such that $N≧(b_0 – a_0)/e$.

I think this is not accurate since it hasn’t showed that such F is Archimedean.. Is such F archimedean? Or in such a condition does there exist such N?

- Can an algebraic extension of an uncountable field be of uncountable degree?
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- Minimum polynomial of $\sqrt{2} + \sqrt{5}$ above $\mathbb{Q}$ (and a generalization)
- Can a field be isomorphic to its subfield but not to a subfield in between?
- Embedding fields into the complex numbers $\mathbb{C}$.

-Definition of a Cauchy sequence;

For every $e\in P_F$, there exists $N\in \omega$ such that if $i,j≧N$, then $|x(i) – x(j)| < e$.

($x:\omega \to F$ is a sequence)

Least Upper Bound Property $\implies$ Complete

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- Question about fields and quotients of polynomial rings
- Show $x^6 + 1.5x^5 + 3x - 4.5$ is irreducible in $\mathbb Q$.
- Polynomials having as roots the sum (respectively, the product) of two algebraic elements
- Construction of $p^n$ field
- Difference between i and -i
- Gaussian Integers and Quotient Rings
- Fixed Field of Automorphisms of $k(x)$

You’re right: there is a problem with the argument, because there is a Cauchy-complete non-Archimedean ordered field, and a non-Archimedean ordered field is not complete in the sense of having least upper bounds.

The standard example starts with the field $F$ of rational functions over $\Bbb R$, with positive cone consisting of those functions $f/g$ such that the leading coefficients of $f$ and $g$ have the same algebraic sign. Then form the Cauchy completion by extending this to equivalence classes of Cauchy sequences in $F$. This is Example 7 on page 17 of Gelbaum & Olmsted, *Counterexample in Analysis*; you may be able to see it here.

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- Useless math that become useful
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