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2. Archimedean property - Wikipedia

   en.wikipedia.org/wiki/Archimedean_property

   The field of the rational numbers endowed with the p-adic metric and the p-adic number fields which are the completions, do not have the Archimedean property as fields with absolute values. All Archimedean valued fields are isometrically isomorphic to a subfield of the complex numbers with a power of the usual absolute value. [6]

3. Absolutely and completely monotonic functions and sequences

   en.wikipedia.org/wiki/Absolutely_and_completely...

   [4] [5] Another related concept is that of a completely/absolutely monotonic sequence. This notion was introduced by Hausdorff in 1921. This notion was introduced by Hausdorff in 1921. The notions of completely and absolutely monotone function/sequence play an important role in several areas of mathematics.

4. Completeness of the real numbers - Wikipedia

   en.wikipedia.org/wiki/Completeness_of_the_real...

   The monotone convergence theorem (described as the fundamental axiom of analysis by Körner [1]) states that every nondecreasing, bounded sequence of real numbers converges. This can be viewed as a special case of the least upper bound property, but it can also be used fairly directly to prove the Cauchy completeness of the real numbers.

5. Absolute value (algebra) - Wikipedia

   en.wikipedia.org/wiki/Absolute_value_(algebra)

   The standard absolute value on the integers. The standard absolute value on the complex numbers.; The p-adic absolute value on the rational numbers.; If R is the field of rational functions over a field F and () is a fixed irreducible polynomial over F, then the following defines an absolute value on R: for () in R define | | to be , where () = () and ((), ()) = = ((), ()).

6. Ostrowski's theorem - Wikipedia

   en.wikipedia.org/wiki/Ostrowski's_theorem

   In number theory, Ostrowski's theorem, due to Alexander Ostrowski (1916), states that every non-trivial absolute value on the rational numbers is equivalent to either the usual real absolute value or a p-adic absolute value.

7. Absolute value - Wikipedia

   en.wikipedia.org/wiki/Absolute_value

   The real absolute value function is an example of a continuous function that achieves a global minimum where the derivative does not exist. The subdifferential of | x | at x = 0 is the interval [−1, 1]. [14] The complex absolute value function is continuous everywhere but complex differentiable nowhere because it violates the Cauchy–Riemann ...

8. Construction of the real numbers - Wikipedia

   en.wikipedia.org/wiki/Construction_of_the_real...

   Substituting a larger value if necessary, we may assume U is rational. Since S is non-empty, we can choose a rational number L such that L < s for some s in S. Now define sequences of rationals (u n) and (l n) as follows: Set u 0 = U and l 0 = L. For each n consider the number m n = (u n + l n)/2. If m n is an upper bound for S, set u n+1 = m n ...

9. Effective results in number theory - Wikipedia

   en.wikipedia.org/wiki/Effective_results_in...

   An early example of an ineffective result was J. E. Littlewood's theorem of 1914, [1] that in the prime number theorem the differences of both ψ(x) and π(x) with their asymptotic estimates change sign infinitely often. [2] In 1933 Stanley Skewes obtained an effective upper bound for the first sign change, [3] now known as Skewes' number.