Math 2.0

definition

Order on the reals

Definition

Construct the real numbers as classes of Cauchy sequences of rational numbers, as in the construction from Cauchy sequences. Let xx and yy be real numbers, represented by the Cauchy sequences (xn)(x_n) and (yn)(y_n) of rational numbers. The notation x<yx < y means that there are a rational number ε>0\varepsilon > 0 and a natural number NN such that yn−xn>εy_n - x_n > \varepsilon for each n≥Nn \ge N. The notation x≤yx \le y means that x<yx < y or x=yx = y.

Remarks

The relation x<yx < y does not depend on the choice of the sequences that represent xx and yy. For all real numbers xx and yy, exactly one of x<yx < y, x=yx = y and y<xy < x holds. Thus ≤\le is a total order. On the rational numbers, this order agrees with the order of the rationals. With Dedekind cuts, the order is inclusion of the lower sets of the cuts.

Explanations

L1 human

Two real numbers are compared through their rational approximations. Suppose that, from some point on, the approximations of yy exceed those of xx by a fixed positive gap. Then xx is less than yy. For example, 1.41,1.414,1.4142,…1.41, 1.414, 1.4142, \dots approximates 2\sqrt{2}, and these approximations are larger than 1.41.4 by more than 0.0050.005, thus 1.4<21.4 < \sqrt{2}.

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Construction of the reals from Cauchy sequences, concept, no evidence levelConstruction of the r…conceptIntegers, object, no evidence levelIntegersobjectOrdered field, definition, no evidence levelOrdered fielddefinitionRational numbers, object, no evidence levelRational numbersobjectRing of integers, statement, no evidence levelRing of integersstatementThe rationals form a field, statement, no evidence levelThe rationals form a …statementThe rationals form an ordered field, statement, no evidence levelThe rationals form an…statementOrder on the reals, definition, no evidence levelOrder on the realsdefinition
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  1. Construction of the reals from Cauchy sequences concept
  2. Integers object
  3. Ordered field definition
  4. Rational numbers object
  5. Ring of integers statement
  6. The rationals form a field statement
  7. The rationals form an ordered field statement
  8. Order on the reals definition

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Sources

  1. Web https://en.wikipedia.org/wiki/Construction_of_the_real_numbers, Construction from Cauchy sequences

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