Glossary

Short definitions of the key terms in §1–§18, written for this site in plain words. They are reminders, not replacements: each entry links to the precise statement in Ross. In the section pages, the first use of a term is underlined with dots. Hover over it or focus it to see its entry here.

A

absolute convergence

A series $\sum a_n$ converges absolutely when $\sum |a_n|$ converges. This is stronger than plain convergence: an absolutely convergent series always converges, but $\sum (-1)^{n+1}/n$ converges without converging absolutely.

In Ross §14 Series

absolute value

The size of a number with its sign dropped: $|a| = a$ when $a \ge 0$ and $|a| = -a$ when $a < 0$. So $|a - b|$ measures how far apart $a$ and $b$ are.

In Ross Definition 3.3 · §3 The Set ℝ of Real Numbers

algebraic number

A number that is a root of some nonconstant polynomial with integer coefficients. Every rational number is algebraic, and so is $\sqrt 2$ (a root of $x^2 - 2$).

In Ross Definition 2.1 · §2 The Set ℚ of Rational Numbers

alternating series

A series whose terms switch sign, like $a_1 - a_2 + a_3 - \cdots$ with every $a_n \ge 0$. If the $a_n$ decrease to $0$, the series converges, and each partial sum is within the next term of the full sum.

In Ross Theorem 15.3 · §15 Alternating Series and Integral Tests

Archimedean property

No real number is larger than every multiple of a fixed positive number: if $a, b > 0$, then $na > b$ for some positive integer $n$. Equivalently, $1/n$ can be made smaller than any given positive number.

In Ross Theorem 4.6 · §4 The Completeness Axiom

B

Bolzano–Weierstrass theorem

Every bounded sequence of real numbers has a convergent subsequence. §13 extends this to bounded sequences in $\mathbb{R}^k$.

In Ross Theorem 11.5, Theorem 13.5 · §11 Subsequences, §13 ★ Some Topological Concepts in Metric Spaces

boundary

The boundary of a set $E$ is made of the points that are near $E$ but not comfortably inside it: points of the closure of $E$ that are not interior points.

In Ross Definition 13.8 · §13 ★ Some Topological Concepts in Metric Spaces

bounded

A set is bounded when it fits inside some interval $[m, M]$. A sequence is bounded when its set of values is, that is, when $|s_n| \le M$ for one constant $M$ and every $n$.

In Ross Definition 4.2 · §4 The Completeness Axiom, §9 Limit Theorems for Sequences

C

Cantor set

What is left of $[0, 1]$ after repeatedly removing the open middle third of every remaining interval. It is closed, contains no interval, has total length $0$, and is still uncountable.

In Ross §13 ★ Some Topological Concepts in Metric Spaces

Cauchy criterion

A series satisfies the Cauchy criterion when its partial sums form a Cauchy sequence: any block of terms far enough out, $a_m + \cdots + a_n$, has small sum. A series converges exactly when it satisfies the criterion.

In Ross Definition 14.3 · §14 Series

Cauchy sequence

A sequence whose terms eventually all lie close to each other: for each $\varepsilon > 0$, all terms past some point are within $\varepsilon$ of one another. In $\mathbb{R}$ these are exactly the convergent sequences, which lets you prove convergence without knowing the limit.

In Ross Definition 10.8, Definition 13.2 · §10 Monotone Sequences and Cauchy Sequences, §13 ★ Some Topological Concepts in Metric Spaces

closed set

A set whose complement is open. Equivalently, a set that contains the limit of every convergent sequence of its points: you cannot escape it by taking limits.

In Ross Definition 13.8 · §13 ★ Some Topological Concepts in Metric Spaces

closure

The smallest closed set containing $E$, written $E^-$. It adds to $E$ every point that is a limit of a sequence in $E$.

In Ross Definition 13.8 · §13 ★ Some Topological Concepts in Metric Spaces

compact

A set is compact when every open cover of it has a finite subcover. In $\mathbb{R}^k$, the Heine–Borel theorem says these are exactly the closed and bounded sets.

In Ross Definition 13.11 · §13 ★ Some Topological Concepts in Metric Spaces

comparison test

A series of nonnegative terms that is term by term smaller than a convergent series converges; one that is term by term larger than a divergent series of nonnegative terms diverges.

In Ross Theorem 14.6 · §14 Series

complete metric space

A metric space in which every Cauchy sequence converges to a point of the space. $\mathbb{R}$ and $\mathbb{R}^k$ are complete; $\mathbb{Q}$ is not.

In Ross Definition 13.2 · §13 ★ Some Topological Concepts in Metric Spaces

Completeness Axiom

Every nonempty set of real numbers that is bounded above has a least upper bound in $\mathbb{R}$. This is the property that separates $\mathbb{R}$ from $\mathbb{Q}$, and almost every existence result in the book rests on it.

In Ross Theorem 4.4 · §4 The Completeness Axiom

continuous

A function $f$ is continuous at $x_0$ when $f(x_n) \to f(x_0)$ for every sequence $x_n \to x_0$ in its domain. Equivalently (ε–δ): $f(x)$ is as close to $f(x_0)$ as you like once $x$ is close enough to $x_0$. Continuous means continuous at every point of the domain.

In Ross Definition 17.1, Theorem 17.2 · §17 Continuous Functions

converges

A sequence $(s_n)$ converges to $s$ when, for every $\varepsilon > 0$, all terms past some index $N$ lie within $\varepsilon$ of $s$. A series converges when its sequence of partial sums does.

In Ross Definition 7.1, Definition 14.2 · §7 Limits of Sequences, §14 Series

D

Dedekind cut

A way to build a real number from rationals alone: a set of rationals that is not empty, not all of $\mathbb{Q}$, closed downward, and with no largest element. Think of it as “all the rationals below the number”.

In Ross §6 ★ A Development of ℝ

denseness of ℚ

Between any two different real numbers there is a rational number. So rationals can approximate any real number as closely as you like.

In Ross Theorem 4.7 · §4 The Completeness Axiom

distance

For real numbers, $\operatorname{dist}(a, b) = |a - b|$. In a metric space the metric plays the same role.

In Ross Definition 3.4 · §3 The Set ℝ of Real Numbers

diverges

Does not converge to any real number. A sequence diverges to $+\infty$ when it eventually exceeds every bound $M$; it can also diverge by oscillating, like $(-1)^n$.

In Ross Definition 7.1, Definition 9.8 · §7 Limits of Sequences, §9 Limit Theorems for Sequences, §14 Series

F

ordered field

A field is a number system with addition and multiplication obeying the usual algebra rules, including division by nonzero numbers. An ordered field also has an order $\le$ that is compatible with them. $\mathbb{Q}$ and $\mathbb{R}$ are ordered fields.

In Ross §3 The Set ℝ of Real Numbers

G

geometric series

A series $\sum a r^n$ in which each term is the previous one times a fixed ratio $r$. With $a \ne 0$, it converges exactly when $|r| < 1$, and then $\sum_{n=0}^\infty a r^n = \dfrac{a}{1 - r}$.

In Ross §14 Series

H

Heine–Borel theorem

A subset of $\mathbb{R}^k$ is compact exactly when it is closed and bounded.

In Ross Theorem 13.12 · §13 ★ Some Topological Concepts in Metric Spaces

I

mathematical induction

A way to prove infinitely many statements $P_1, P_2, \dots$ at once: prove $P_1$ (the basis), then prove that each $P_n$ implies $P_{n+1}$ (the induction step).

In Ross §1 The Set ℕ of Natural Numbers

infimum

The greatest lower bound of a set, written $\inf S$: a lower bound that is at least as large as every other lower bound. It need not belong to the set.

In Ross Definition 4.3 · §4 The Completeness Axiom, §5 The Symbols +∞ and −∞

integral test

For a series of positive, decreasing terms $a_n = f(n)$, compare $\sum a_n$ with $\int_1^\infty f(x)\,dx$: both converge or both diverge. It is how one shows that $\sum 1/n^p$ converges exactly when $p > 1$.

In Ross §15 Alternating Series and Integral Tests

interior point

A point $s_0$ is interior to $E$ when some small ball around $s_0$ lies entirely inside $E$. The interior $E^\circ$ is the set of all such points.

In Ross Definition 13.6 · §13 ★ Some Topological Concepts in Metric Spaces

Intermediate Value Theorem

A continuous function on an interval takes every value between any two of its values. So a continuous $f$ with $f(a) < 0 < f(b)$ has a root between $a$ and $b$.

In Ross Theorem 18.2 · §18 Properties of Continuous Functions

interval

$[a, b]$ is the closed interval of points between $a$ and $b$ including both ends; $(a, b)$ is the open interval, without the ends; $[a, b)$ and $(a, b]$ are half-open.

In Ross §4 The Completeness Axiom

irrational number

A real number that is not rational, such as $\sqrt 2$ or $\pi$. Irrational numbers are also dense: there is one between any two reals.

In Ross §2 The Set ℚ of Rational Numbers, §4 The Completeness Axiom

L

limit

The number $s$ that a convergent sequence approaches, written $\lim s_n = s$ or $s_n \to s$. A sequence has at most one limit.

In Ross Definition 7.1 · §7 Limits of Sequences

lim sup and lim inf

$\limsup s_n$ is the limit of the suprema of the tails $\{s_n : n > N\}$; it is the largest value that terms of the sequence keep coming back near. $\liminf s_n$ is the same with infima. They always exist (possibly $\pm\infty$), and they are equal exactly when $\lim s_n$ exists.

In Ross Definition 10.6 · §10 Monotone Sequences and Cauchy Sequences, §12 lim sup's and lim inf's

lower bound

A number $m$ with $m \le s$ for every $s$ in $S$. A set with a lower bound is bounded below.

In Ross Definition 4.2 · §4 The Completeness Axiom

M

maximum and minimum

The largest element of a set, if the set has one, and likewise the smallest. $(0, 1)$ has neither, even though it has a supremum and an infimum.

In Ross Definition 4.1 · §4 The Completeness Axiom

metric space

A set together with a metric: a distance function $d$ that is zero only from a point to itself, symmetric, and satisfies the triangle inequality. Limits, open sets and compactness all make sense in any metric space.

In Ross Definition 13.1 · §13 ★ Some Topological Concepts in Metric Spaces

monotone (increasing, decreasing)

A sequence is increasing if $s_n \le s_{n+1}$ for all $n$, decreasing if $s_n \ge s_{n+1}$ for all $n$, and monotone if it is one of these. A bounded monotone sequence always converges. A function is increasing when $x < y$ implies $f(x) \le f(y)$, and strictly increasing when it implies $f(x) < f(y)$.

In Ross Definition 10.1 · §10 Monotone Sequences and Cauchy Sequences

O

open cover

A family of open sets whose union contains $E$. A subcover is part of the family that still contains $E$.

In Ross Definition 13.11 · §13 ★ Some Topological Concepts in Metric Spaces

open set

A set in which every point is an interior point: each point has a small ball around it that stays in the set.

In Ross Definition 13.6 · §13 ★ Some Topological Concepts in Metric Spaces

P

partial sum

The sum of the first terms of a series, $s_n = a_m + \cdots + a_n$. A series is defined to converge to the limit of its partial sums.

In Ross Definition 14.2 · §14 Series

Peano axioms

The five basic properties of the natural numbers $\mathbb{N}$: $1$ is in $\mathbb{N}$, every $n$ has a successor $n + 1$, and so on. The last one, N5, is the principle behind mathematical induction.

In Ross §1 The Set ℕ of Natural Numbers

R

ratio test

Look at $|a_{n+1}/a_n|$. If its lim sup is below $1$, $\sum a_n$ converges absolutely; if its lim inf is above $1$, the series diverges. Otherwise the test says nothing.

In Ross Theorem 14.8 · §14 Series

rational number

A number $m/n$ with $m, n$ integers and $n \ne 0$. The rationals $\mathbb{Q}$ form an ordered field, but they have gaps: $\sqrt 2$ is not one of them.

In Ross §2 The Set ℚ of Rational Numbers

Rational Zeros Theorem

If $c/d$ in lowest terms is a root of a polynomial with integer coefficients, then $c$ divides the constant term and $d$ divides the leading coefficient. It gives a short list of candidates for rational roots, and shows numbers like $\sqrt 2$ are irrational.

In Ross Theorem 2.2 · §2 The Set ℚ of Rational Numbers

repeating decimal

A decimal expansion that ends in one block of digits repeated forever, such as $0.1\overline{36}$. A real number is rational exactly when its decimal expansion repeats.

In Ross Definition 16.4 · §16 ★ Decimal Expansions of Real Numbers

root test

Look at $\limsup |a_n|^{1/n}$. If it is below $1$, $\sum a_n$ converges absolutely; if it is above $1$, the series diverges; if it equals $1$, the test says nothing.

In Ross Theorem 14.9 · §14 Series

S

sequence

A list $s_1, s_2, s_3, \dots$ of numbers indexed by integers, written $(s_n)$. Formally, a function on a set of integers $\{n : n \ge m\}$. Its terms can repeat, and their order matters.

In Ross §7 Limits of Sequences

series

An infinite sum $\sum a_n = a_m + a_{m+1} + \cdots$. Its value, when it has one, is the limit of its partial sums.

In Ross Definition 14.2 · §14 Series

squeeze lemma

If $a_n \le s_n \le b_n$ for all $n$ and $a_n$ and $b_n$ have the same limit $s$, then $s_n$ converges to $s$ too.

In Ross Exercise 8.5 · §8 A Discussion about Proofs

subsequence

A sequence formed by picking terms $s_{n_1}, s_{n_2}, s_{n_3}, \dots$ of $(s_n)$ with $n_1 < n_2 < n_3 < \cdots$: some terms are skipped, the rest keep their order.

In Ross Definition 11.1 · §11 Subsequences

subsequential limit

A limit (possibly $\pm\infty$) of some subsequence. For example, $(-1)^n$ has the two subsequential limits $1$ and $-1$.

In Ross Definition 11.6 · §11 Subsequences

supremum

The least upper bound of a set, written $\sup S$: an upper bound that is no larger than any other upper bound. It need not belong to the set; $\sup (0, 1) = 1$.

In Ross Definition 4.3 · §4 The Completeness Axiom, §5 The Symbols +∞ and −∞

T

triangle inequality

$|a + b| \le |a| + |b|$, or in terms of distance, $\operatorname{dist}(a, c) \le \operatorname{dist}(a, b) + \operatorname{dist}(b, c)$: going through $b$ is never shorter. It is the workhorse of ε-arguments.

In Ross Theorem 3.7 · §3 The Set ℝ of Real Numbers, §13 ★ Some Topological Concepts in Metric Spaces

U

uncountable

A set is countable when its elements can be listed as a sequence; otherwise it is uncountable. $\mathbb{Q}$ is countable, but $\mathbb{R}$, any interval, and the Cantor set are uncountable.

In Ross §13 ★ Some Topological Concepts in Metric Spaces, §16 ★ Decimal Expansions of Real Numbers

upper bound

A number $M$ with $s \le M$ for every $s$ in $S$. A set with an upper bound is bounded above.

In Ross Definition 4.2 · §4 The Completeness Axiom