The Real Line: Standing Notation and Background for Calculus

This setting fixes notation and background facts about the real numbers for items that adopt it. It introduces no new concept, and every fact recorded here is carried by reference to the item that establishes it.

1. The real field. ¶ $\mathbb{R}$ denotes the real numbers: an ordered field in which every nonempty subset bounded above has a least upper bound. Its order is written $\le$, and $<$ denotes the associated strict order, so that $s<t$ means $s\le t$ and $s\ne t$. Addition, multiplication, the identities $0$ and $1$, and additive and multiplicative inverses are those of the underlying field; we write $s-t$ for $s+(-t)$ and, when $t\ne 0$, $s/t$ for $s\,t^{-1}$. The elementary consequences of the field and order axioms collected in Elementary Arithmetic in an Ordered Field and Elementary Order Arithmetic in an Ordered Field are used without further comment, as is the nonnegativity of squares recorded in Nonnegativity of Squares in an Ordered Field.

2. Suprema and infima. ¶ Upper bounds and least upper bounds are as in Upper Bound and Least Upper Bound, and lower bounds and greatest lower bounds as in Lower Bound and Greatest Lower Bound in a Totally Ordered Set, for the total order $\le$ of $\mathbb{R}$. By Uniqueness of the Supremum and of the Infimum each is unique when it exists, and they are written $\sup S$ and $\inf S$. Every nonempty subset of $\mathbb{R}$ bounded above has a supremum, by the least upper bound property of The Real Numbers; every nonempty subset of $\mathbb{R}$ bounded below has an infimum, by Existence of the Infimum of a Nonempty Subset of $\mathbb{R}$ Bounded Below. The approximation property of Approximation Property of the Supremum and the Infimum in $\mathbb{R}$ is used freely.

3. Absolute value and the real line. ¶ For $s\in\mathbb{R}$, $|s|$ denotes the absolute value of $s$, with the properties recorded in Properties of the Absolute Value in an Ordered Field. The real line is the metric space $(\mathbb{R},d_{\mathbb{R}})$ with $d_{\mathbb{R}}(s,t)=|s-t|$, as established in The Absolute Value Metric on the Real Line. Every subset of $\mathbb{R}$ occurring below is regarded as a subset of this metric space, and $\mathbb{R}$ as a codomain always carries $d_{\mathbb{R}}$.

4. Intervals. ¶ A subset $I\subseteq\mathbb{R}$ is an interval if $y\in I$ whenever $x,z\in I$ and $x\le y\le z$. For $a\le b$ the closed interval $[a,b]$ is $\{x\in\mathbb{R}:a\le x\le b\}$, as in that definition, and for $p,q\in\mathbb{R}$ the open interval $(p,q)$ is $\{x\in\mathbb{R}:p<x\text{ and }x<q\}$. A point $x\in I$ is an interior point of $I$ if $u<x<v$ for some $u,v\in I$. The routine facts about intervals established in Basic Facts about Intervals of the Real Line and Their Interior Points are used throughout and without further comment: $\mathbb{R}$ is itself an interval containing at least two points, every real number being an interior point of it; if $u,v\in I$ with $u\le v$ then $[u,v]\subseteq I$; a point lying strictly between two points of $I$ belongs to $I$ and is an interior point of $I$; every open interval $(p,q)$ is an interval each of whose points is an interior point of it; and every closed interval $[a,b]$ with $a\le b$ is an interval, which when $a<b$ contains at least two points and has every point of $(a,b)$ among its interior points.

5. Continuity. ¶ Let $E\subseteq\mathbb{R}$, let $f:E\to\mathbb{R}$ and let $x\in E$. The phrases $f$ is continuous at $x$ and $f$ is continuous on $E$ always mean, respectively, that $f$ is continuous at $x$ relative to $E$ and continuous on $E$, for $f$ regarded as a map from the subset $E$ of the real line into the real line. Written out through the absolute value metric of clause 3, continuity of $f$ at $x$ says: for every $\varepsilon>0$ there exists $\delta>0$ such that every $y\in E$ with $|y-x|<\delta$ satisfies $|f(y)-f(x)|<\varepsilon$. Sums, products and constant multiples of continuous real-valued functions are continuous by Continuity of Sums and Products of Real-Valued Functions on a Metric Space, and a restriction of a continuous function to a smaller subset is continuous by clause 1 of Restriction Stability of Continuity and of the Derivative. The identity map of $E$, every natural-number power map on $E$, and the restriction to $E$ of any polynomial function on $\mathbb{R}$ are continuous on $E$, by Continuity of the Identity Map, of Powers, and of Polynomial Functions on a Subset of the Real Line. Uniform continuity on $E$ is as in Uniform Continuity on a Subset of the Real Numbers.

6. Natural numbers and induction. ¶ $\mathbb{N}$ denotes the set of natural numbers, subject to the principle of induction, and each $n\in\mathbb{N}$ is identified with its image in $\mathbb{R}$ under the canonical map.

7. Sequences. ¶ A sequence of real numbers is a sequence in $\mathbb{R}$, written $(a_n)_{n\in\mathbb{N}}$, or equivalently $(a_n)_{n=1}^{\infty}$, which is the notation used by some of the items cited below. It converges to $L\in\mathbb{R}$ if for every $\varepsilon>0$ there is $N\in\mathbb{N}$ such that $|a_n-L|<\varepsilon$ for every $n\ge N$; such an $L$ is unique by clause 1 of Uniqueness of Limits and Boundedness of Convergent Real Sequences, and one writes $\lim_{n\to\infty}a_n=L$. Boundedness of a sequence is as in Bounded Sequence of Real Numbers. The arithmetic rules for limits of Arithmetic of Limits of Real Sequences and the order properties of Order Properties of Limits of Real Sequences are used freely, as are the convergence of a constant sequence to its value and the invariance of the limit under a shift of the index, both established in Constant Sequences and Index-Shifted Sequences of Real Numbers.

8. Square roots. ¶ For $\alpha\in\mathbb{R}$ with $0\le\alpha$ we write $\sqrt{\alpha}$ for the unique real number $r$ satisfying $0\le r$ and $r^2=\alpha$; such an $r$ exists and is unique by Existence and Uniqueness of the Nonnegative Square Root. For nonnegative real numbers an inequality holds between two of them if and only if it holds between their squares, by Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field.