The Real Number Sequence

AKA: R, Real-Number Line.
Context:
- It includes the Rational Number Sequence.
- It includes the Irrational Number Sequence.
- It can be represented by:
  - the Symbol [math]\displaystyle{ \mathbb{R} }[/math] (\mathbb{R});
  - or, the Numeric Interval [math]\displaystyle{ (-\infty, \infty)_\mathbb{R} }[/math].
- It can be a Supersequence to a Real Number Sequence.
- …
Counter-Example(s):
- The Complex Number Sequence.
- The Integer Sequence.
See: Real Number-Output Function.

References

(Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/real_line Retrieved:2015-6-22.
- In mathematics, the real line, or real number line is the line whose points are the real numbers. That is, the real line is the set $R$ of all real numbers, viewed as a geometric space, namely the Euclidean space of dimension one. It can be thought of as a vector space (or affine space), a metric space, a topological space, a measure space, or a linear continuum.
  Just like the set of real numbers, the real line is usually denoted by the symbol $R$ (or alternatively, [math]\displaystyle{ \mathbb{R} }[/math], the letter “R” in blackboard bold). However, it is sometimes denoted $R 1$ in order to emphasize its role as the first Euclidean space.
  This article focuses on the aspects of $R$ as a geometric space in topology, geometry, and real analysis. The real numbers also play an important role in algebra as a field, but in this context R is rarely referred to as a line. For more information on R in all of its guises, see real number.

(Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/real_number#Definition Retrieved:2015-6-22.
- The real number system [math]\displaystyle{ (\mathbb R ; + ; \cdot ; \lt ) }[/math] can be defined axiomatically up to an isomorphism, which is described hereafter. There are also many ways to construct "the" real number system, for example, starting from natural numbers, then defining rational numbers algebraically, and finally defining real numbers as equivalence classes of their Cauchy sequences or as Dedekind cuts, which are certain subsets of rational numbers. Another possibility is to start from some rigorous axiomatization of Euclidean geometry (Hilbert, Tarski, etc.) and then define the real number system geometrically. From the structuralist point of view all these constructions are on equal footing.

(Wikipedia, 2009) ⇒ http://en.wikipedia.org/wiki/Real_number#Definition
- Let R denote the set of all real numbers. Then:
  - The set R is a field, meaning that addition and multiplication are defined and have the usual properties.
  - The field R is ordered, meaning that there is a total order ≥ such that, for all real numbers x, y and z:
    - if x ≥ y then x + z ≥ y + z;
    - if x ≥ 0 and y ≥ 0 then xy ≥ 0.
  - The order is Dedekind-complete; that is, every non-empty subset S of R with an upper bound in R has a least upper bound (also called supremum) in R.
- The last property is what differentiates the reals from the rationals. For example, the set of rationals with square less than 2 has a rational upper bound (e.g., 1.5) but no rational least upper bound, because the square root of 2 is not rational.
(Wikipedia, 2009) ⇒ http://en.wikipedia.org/wiki/Construction_of_the_real_numbers
- In mathematics, there are several ways of defining the real number system as an ordered field. The synthetic approach gives a list of axioms for the real numbers as a complete ordered field. Under the usual axioms of set theory, one can show that these axioms are categorical, in the sense that there is a model for the axioms, and any two such models are isomorphic. Any one of these models must be explicitly constructed, and most of these models are built using the basic properties of the rational number system as an ordered field.