The Real Numbers: Standing Notation and Background

This setting fixes the standing notation for the real numbers, their sequences and their bounds, used by results that require no further structure. It introduces no new concepts.

1. (Numbers)(i) $\mathbb{R}$ is the ordered field of real numbers, with the notation of that item: the order $\le$, the strict order $<$, the words positive, nonnegative, negative and nonpositive, the difference $s-t$, and the quotient $\tfrac{s}{t}=s\,t^{-1}$ of $s$ by a nonzero $t$. Natural numbers are read in $\mathbb{R}$ through the canonical map, whose properties are those of Properties of the Canonical Map from the Natural Numbers to an Ordered Field; such a number is positive and invertible by claim 3 of that lemma, so that $\tfrac{s}{n}$ is defined for every $n\in\mathbb{N}$. For $s\in\mathbb{R}$, $|s|$ is the absolute value of $s$, and $d_{\mathbb{R}}(s,t)=|s-t|$ is the metric on $\mathbb{R}$ of The Absolute Value Metric on the Real Numbers, so that $(\mathbb{R},d_{\mathbb{R}})$ is a metric space; $s^{n}$ is the natural power of $s$, and $s^{2}=ss$.

2. (Natural numbers and finite sums)(ii) $\mathbb{N}$ is the set of natural numbers, with successor map $S$ as in that item and the order of that definition, whose properties are those of Properties of the Order on the Natural Numbers; $[n]$ is the initial segment of $n\in\mathbb{N}$, and $X^{n}$ the set of $n$-tuples in a set $X$. Finite sums of real numbers are the finite sums in a field; when the summands are the values of a map defined on a set containing $[n]$, the finite sum $\sum_{k=1}^{n}a_{k}$ is that of the restriction of the map to $[n]$, and by claim 1 of Properties of Finite Sums it does not depend on which such map is used. The decimal numerals $2,3,4,\dots$ name the natural numbers $S(1),S(2),S(3),\dots$ in order. Read in $\mathbb{R}$ as in clause 1, consecutive numerals differ by $1$, so that $2=1+1$, $3=2+1$, and so on, by claim 1 of Arithmetic of Addition on the Natural Numbers and claim 1 of Properties of the Canonical Map from the Natural Numbers to an Ordered Field.

3. (Integers and sums over finite sets)(iii) $\mathbb{Z}\subseteq\mathbb{R}$ is the set of integers, with the order of $\mathbb{R}$. For a nonempty finite set $G$ and a map $f:G\to\mathbb{R}$, $\sum_{x\in G}f(x)$ is the sum over the finite index set $G$; for a nonempty subset $E\subseteq G$, which is finite by claim 3 of Basic Properties of Finite Sets, $\sum_{x\in E}f(x)$ is the sum of the restriction of $f$ to $E$. For $G=[n]$ it agrees with $\sum_{k=1}^{n}f(k)$ by claim 1 of Properties of a Sum over a Finite Index Set.

4. (Sequences)(iv) A sequence is indexed by $\mathbb{N}$, and its subsequences are as defined there, with Strictly Increasing Sequences of Natural Numbers Dominate Their Index in force. A sequence of real numbers converges to a real number, and is a Cauchy sequence, as defined there; by Convergence, Limits and the Cauchy Condition for Real Sequences Agree with Those in the Real Line as a Metric Space, which is Complete these agree with convergence and with the Cauchy condition in $(\mathbb{R},d_{\mathbb{R}})$, and every Cauchy sequence of real numbers converges by Every Cauchy Sequence of Real Numbers Converges. The limit of a convergent sequence is unique by claim 1 of Uniqueness of Limits and Boundedness of Convergent Real Sequences, and is written $\lim_{m\to\infty}a_{m}$.

5. (Bounds)(v) Upper and lower bounds of a subset of $\mathbb{R}$, and least upper and greatest lower bounds, are as in Upper Bound and Least Upper Bound and Lower Bound and Greatest Lower Bound in a Totally Ordered Set. A nonempty $S\subseteq\mathbb{R}$ that is bounded above has a least upper bound, since $\mathbb{R}$ is Dedekind complete, and it is unique by Uniqueness of the Supremum and of the Infimum; it is written $\sup S$. A nonempty $S\subseteq\mathbb{R}$ that is bounded below has a unique greatest lower bound $\inf S$ by Existence of the Infimum of a Nonempty Subset of $\mathbb{R}$ Bounded Below. For a nonempty set $A$ and a map $u:A\to\mathbb{R}$ we write $u(A)=\{u(x):x\in A\}$ and call $u$ bounded above, respectively bounded below, if $u(A)$ is; in those cases $\sup_{x\in A}u(x)$ denotes $\sup u(A)$ and $\inf_{x\in A}u(x)$ denotes $\inf u(A)$.

6. (Background)(vi) The following results are in force by reference: Elementary Order Arithmetic in an Ordered Field, Elementary Arithmetic in an Ordered Field, Properties of the Absolute Value in an Ordered Field, Properties of Natural Number Powers in a Field, Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field, Existence and Uniqueness of the Nonnegative Square Root, Properties of Finite Sums, Approximation Property of the Supremum and the Infimum in $\mathbb{R}$, The Archimedean Property of the Real Numbers, Arithmetic of Limits of Real Sequences, Order Properties of Limits of Real Sequences, Uniqueness of Limits and Boundedness of Convergent Real Sequences and Comparison of Real Numbers with Arbitrary Positive Slack; for field arithmetic, squares and maxima, Zero Products and Elementary Identities in a Field, Nonnegativity of Squares in an Ordered Field and Elementary Properties of the Maximum of Two Elements; for natural numbers and integers, Arithmetic of Addition on the Natural Numbers, Properties of the Order on the Natural Numbers, Properties of the Canonical Map from the Natural Numbers to an Ordered Field and Arithmetic, Order, Discreteness and Intervals of the Integers; for finite sets and finite sums, Basic Properties of Finite Sets, Finite Sums of Real Numbers: Nonnegativity, Domination by the Sum, and Limits, Comparison and Absolute Value Bounds for Finite Sums of Real Numbers, Properties of a Sum over a Finite Index Set, Peeling, Splitting, and Interchange for Sums over a Finite Index Set, Sums over Finite Index Sets: Finite Unions, Disjoint Unions, Vanishing Terms, Dependent Pairs, Conjugation and the Modulus and Real Sums over a Finite Index Set: Comparison, Nonnegativity, Monotonicity, Term Bounds, Absolute Values, Counting and Limits.