
Abel's convergence test, 247

Abel's theorem for power series, 249, 322

Absolute value

in an ordered field, 26

in E^{n}, 64

in Euclidean spaces, 88

in normed linear spaces, 90

Absolutely continuous functions (weakly), 309

Absolutely convergent series of functions, 237

rearrangement of, 238

tests for, 239

Accumulation points, 115. See also Cluster point

Additivity

of definite integrals, 282

of total variation, 301

of volume of intervals in E^{n}, 79

Alternating series, 248

Admissible change of variable, 165

Angle between vectors in E^{n}, 70

Antiderivative, 278. See also Integral, indefinite

Antidifferentiation, 278. See also Integration

Arcs, 211

as connected sets, 214

endpoints of, 211

length of, 301, 311

rectifiable, 309

simple, 211

Archimedean field, see Field, Archimedean

Archimedean property, 43

Arcwise connected set, 211

Arithmeticgeometric mean, Gauss's, 134

Associative laws

in a field, 23

of vector addition in E^{n}, 65

Axioms

of arithmetic in a field, 23

of a metric, 95

of order in an ordered field, 24

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Basic unit vector in E^{n}, 64

Bernoulli inequalities, 33

Binary operations, 12. See also Functions

Binomial theorem, 34

Bolzano theorem, 205

BolzanoWeierstrass theorem, 136

Boundary

of intervals in E^{n}, 77

of sets in metric spaces, 108

Bounded

functions on sets in metric spaces, 111

sequences in metric spaces, 111

sets in metric spaces, 109

sets in ordered fields, 36

variation, 303

leftbounded sets in ordered fields, 36

rightbounded sets in ordered fields, 36

totally bounded sets in a metric space, 188

uniformly bounded sequences of functions, 234

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C (the complex field), 80

complex numbers, 81; see also Complex numbers

Cartesian coordinates in, 83

de Moivre's formula, 84

imaginary numbers in, 81

imaginary unit in, 81

is not an ordered field, 82

polar coordinates in, 83

real points in, 81

real unit in, 81

C^{n} (complex nspace), 87

as a Euclidean space, 88

as a normed linear space, 91

componentwise convergence of sequences in, 121

dot products in, 87

standard norm in, 91

Cantor's diagonal process, 21. See also Sets

Cantor's function, 186

Cantor's principle of nested closed sets, 188

Cantor's set, 120

Cartesian coordinates in C, 83

Cartesian product of sets, 2

intervals in E^{n} as Cartesian products of intervals in E^{1}, 76

Cauchy criterion

for function limits, 162

for uniform convergence of sequences of functions, 231

Cauchy form of the remainder term of Taylor expansions, 291

Cauchy sequences in metric spaces, 141

Cauchy's convergence criterion for sequences in metric spaces, 143

Cauchy's laws of the mean, 261

CauchySchwarz inequality

in E^{n}, 67

in Euclidean spaces, 88

Center of an interval in E^{n}, 77

Change of variable, admissible, 165

Chain rule for differentiation of composite functions, 255

Change of variables in definite integrals, 282

Characteristic functions of sets, 323

Clopen

sets in metric spaces, 103

Closed

curve, 211

globe in a metric space, 97

interval in an ordered field, 37

interval in E^{n}, 77

line segment in E^{n}, 72

sets in metric spaces, 103, 138

Closures of sets in metric spaces, 137

Closure laws

in a field, 23

in E^{n}, 65

of integers in a field, 35

of rationals in a field, 35

Cluster points

of sequences in E^{*}, 60

of sequences and sets in metric spaces, 115

Commutative laws

in a field, 23

of addition of vectors in E^{n}, 65

of inner products of vectors in E^{n}, 67

Compact sets, 186, 193

Cantor's principle of nested closed sets, 188

are totally bounded, 188

in E^{1}, 195

continuity on, 194

generalized HeineBorel theorem, 193

HeineBorel theorem, 324

sequentially, 186

Comparison test, 239

refined, 245

Complement of a set, 2

Complete

metric spaces, 143

ordered fields, 38; see also Field, complete ordered

Completeness axiom, 38

Completion of metric spaces, 146

Complex exponential, 173

derivatives of the, 256

Complex field, seeC

Complex functions, 170

Complex numbers, 81. See also C

conjugate of, 81

imaginary part of, 81
 nth roots of, 85

polar form of, 83

real part of, 81

trigonometric form of, 83

Complex vector spaces, 87

Componentwise

continuity of functions, 172

convergence of sequences, 121

differentiation, 256

integration, 282

limits of functions, 172

Composite functions, 163

chain rule for derivatives of, 255

continuity of, 163

Concurrent sequences, 144

Conditionally convergent series of functions, 237

rearrangement of, 250

Conjugate of complex numbers, 81

Connected sets, 212

arcs as, 214

arcwise, 211

curves as, 214

polygon, 204

Continuous functions

on metric spaces, 149

differentiable functions are, 252

left, 153

relatively, 152

right, 153

uniformly, 197

weakly absolutely continuous, 309

Continuity. See also Continuous functions

componentwise, 172

in one variable, 174

jointly, 174

of addition and multiplication in E^{1}, 168

of composite functions, 163

of inverse functions, 195, 207

of the exponential function, 184

of the logarithmic function, 208

of the power function, 209

of the standard metric on E^{1}, 168

of the sum, product, and quotient of functions, 170

on compact sets, 194

sequential criterion for, 161

uniform, 197

Contracting sequence of sets, 17

Contraction mapping, 198

Convergence of sequences of functions

Cauchy criterion for uniform, 231

convergence of integrals and derivatives, 315

pointwise, 228

uniform, 228

Convergence radius of power series, 243

Convergence tests for series

Abel's test, 247

comparison test, 239

Dirichlet test, 248

integral test, 327

Leibniz test for alternating series, 248

ratio test, 241

refined comparison test, 245

root test, 241

Weierstrass Mtest for functions, 240

Convergent

absolutely convergent series of functions, 237

conditionally convergent series of functions, 237

sequences of functions, 228; see also Limits of sequences of functions

sequences in metric spaces, 115

series of functions, 228; see also Limits of series of functions

Convex sets, 204

piecewise, 204

Coordinate equations of a line in E^{n}, 72

Countable set, 18

rational numbers as a, 19

Countable union of sets, 20

Covering, open, 192

Cross product of sets, 2

Curves, 211

as connected sets, 214

closed, 211

length of, 300

parametric equations of, 212

tangent to, 257

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Darboux property, 203

Bolzano theorem, 205

of the derivative, 265

de Moivre's formula, 84

Definite integrals, 279

additivity of, 282

change of variables in, 282

dominance law for, 284

first law of the mean for, 285

integration by parts, 281

linearity of, 280

monotonicity law for, 284

weighted law of the mean for, 286, 326

Degenerate intervals in E^{n}, 78

Degree

of a monomial, 173

of a polynomial, 173

Deleted globes about points in metric spaces, 150

Dense subsets in metric spaces, 139

Density

of an ordered field, 45

of rationals in an Archimedean field, 45

Dependent vectors

in E^{n}, 69

Derivatives of functions on E^{1}, 251

convergence of, 315

Darboux property of, 265

derivative of the exponential function, 264

derivative of the inverse function, 263

derivative of the logarithmic function, 263

derivative of the power function, 264

with extendedreal values, 259

left, 252

onesided, 252

right, 252

Derived functions on E^{1}, 251
 nth, 252

Diagonal of an interval in E^{n}, 77

Diagonal process, Cantor's, 21. See also Sets

Diameter

of sets in metric spaces, 109

Difference

of elements of a field, 26

of sets, 2

Differentials of functions on E^{1}, 288

of order n, 289

Differentiable functions on E^{1}, 251

Cauchy's laws of the mean, 261

cosine function, 337

are continuous, 252

exponential function, 333

infinitely, 292

logarithmic function, 332

ntimes continuously, 292

ntimes, 252

nowhere, 253

Rolle's theorem, 261

sine function, 337

Differentiation, 251

chain rule for, 255

componentwise, 256

of power series, 319

rules for sums, products, and quotients, 256

termwise differentiation of series, 318

Directed

lines in E^{n}, 74

planes in E^{n}, 74

Direction vectors
of lines in E^{n}, 71

Dirichlet function, 155, 329

Dirichlet test, 248

Disconnected sets, 212

totally, 217

Discontinuity points of functions on metric spaces, 149

Discontinuous functions on metric spaces, 149

Discrete

metric, 96

metric space, 96

Disjoint sets, 2

Distance

between a point and a plane in E^{n}, 76

between sets in metric spaces, 110

between two vectors in E^{n}, 64

between two vectors in Euclidean spaces, 89

in normed linear spaces, 92

norminduced, 92

translationinvariant, 93

Distributive laws

in E^{n}, 65

in a field, 24

of inner products of vectors in E^{n},
67

of union and intersection of sets, 7

Divergent

sequences in metric spaces, 115

Domain

of a relation, 9

of a sequence, 15

space of functions on metric spaces, 149

Double limits of functions, 219, 221

Double sequence, 20, 222, 223

Dot product

in C^{n}, 87

in E^{n}, 64

Duality laws, de Morgan's, 3. See also Sets

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e (the number), 122, 165, 293

E^{1} (the real numbers), 23. See alsoField, complete ordered

associative laws in, 23

axioms of arithmetic in, 23

axioms of order in, 24

closure laws in, 23

commutative laws in, 23

continuity of addition and multiplication in, 168

continuity of the standard metric on, 168

distributive law in, 24

inverse elements in, 24

monotonicity in, 24

neighborhood of a point in, 58

natural numbers in, 28

neutral elements in, 23

transitivity in, 24

trichotomy in, 24

E^{n} (Euclidean nspace), 63. See also Vectors in E^{n}

convex sets in, 204

as a Euclidean space, 88

as a normed linear space, 91

associativity of vector addition in, 65

additive inverses of vector addition, 65

basic unit vector in, 64

BolzanoWeierstrass theorem, 136

CauchySchwarz inequality in, 67

closure laws in, 65

commutativity of vector addition in, 65

componentwise convergence of sequences in, 121

distributive laws in, 65

globe in, 76

hyperplanes in, 72; see also Planes in E^{n}

intervals in, 76; see also Intervals in E^{n}

line segments in, 72; see also Line segments in E^{n}

linear functionals on, 74, 75; see also Linear functionals on E^{n}

lines in, 71; see also Lines in E^{n}

neutral element of vector addition in, 65

planes in, 72; see also Planes in E^{n}

point in, 63

scalar of, 64

scalar product in, 64

sphere in, 76

standard metric in, 96

standard norm in, 91

triangle inequality of the absolute value in, 67

triangle inequality of the distance in, 68

unit vector in, 65

vectors in, 63

zero vector in, 63

E^{*} (extended real numbers), 53

as a metric space, 98

cluster point of a sequence in, 60

globes in, 98

indeterminate expressions in, 178

intervals in, 54

limits of sequences in, 58

metrics for, 99

neighborhood of a point in, 58

operations in, 177

unorthodox operations in, 180

Edgelengths of an interval in E^{n}, 77

Elements of a set, 1

Empty set, 1

Endpoints

of an interval in E^{n}, 77

of line segments in E^{n}, 72

Equality of sets, 1

Equicontinuous functions, 236

Equivalence class relative to an equivalence relation, 13

generator of an, 13

representative of an, 13

Equivalence relation, 12

equivalence class relative to an, 13

Euclidean nspace, see E^{n}

Euclidean spaces, 88

as normed linear spaces, 91

absolute value in, 88

C^{n} as a, 88

CauchySchwarz inequality in, 88

distance in, 89

E^{n} as a, 88

line segments in, 89

lines in, 89

planes in, 89

triangle inequality in, 88

Exact primitive, 278

Existential quantifier, 4

Expanding sequence of sets, 17

Exponential, complex, 173

Exponential function, 183, 333

continuity of the, 184

derivative of the, 264

inverse of the, 208

Extended real numbers, see E^{*}

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Factorials, definition of, 31

Family of sets, 3

intersection of a, 3

union of a, 3

Fields, 25

associative laws in, 23

axioms of arithmetic in, 23

binomial theorem, 34

closure laws in, 23

commutative laws in, 23

difference of elements in, 26

distributive law in, 24

first induction law in, 28

inductive definitions in, 31

inductive sets in, 28

integers in, 34

inverse elements in, 24

irrationals in, 34

Lagrange identity in, 71

natural elements in, 28

neutral elements in, 23

quotients of elements in, 26

rational subfields of, 35

rationals in, 34

Fields, Archimedean, 43. See also Fields, ordered

density of rationals in, 45

integral parts of elements of, 44

Fields, complete ordered, 38. See also Field, Archimedean

Archimedean property of, 43

completeness axiom, 38

density of irrationals in, 51

existence of irrationals in, 46

powers with rational exponents in, 47

powers with real exponents in, 50

principle of nested intervals in, 42

roots in, 46

Fields, ordered, 25. See also Field

absolute value in, 26

axioms of order in, 24

Bernoulli inequalities in, 33

bounded sets in, 36

closed intervals in, 37

density of, 45

greatest lower bound (glb) of sets in, 38

halfclosed intervals in, 37

halfopen intervals in, 37

infimum (inf) of sets in, 38

intervals in, 37

least upper bound (lub) of sets in, 37

monotonicity in, 24

negative elements in, 25

open intervals in, 37

positive elements in, 25

rational subfield in, 35

second induction law in, 30

supremum (sup) of sets in, 38

transitivity in, 24

trichotomy in, 24

wellordering of naturals in, 30

Finite

increments law, 271

intervals, 54

sequence, 16

set, 18

First

induction law, 28

law of the mean, 285

Functions, 10. See also Functions on E^{1} and Functions on metric spaces

binary operations, 12

bounded, 96

Cantor's function, 186

characteristic, 323

complex, 170

Dirichlet function, 155, 329

equicontinuous, 236

graphs of, 153

isometry, 201

limits of sequences of, see Limits of sequences of functions

limits of series of, see Limits of series of functions

monotone, 181

nondecreasing, 181

nonincreasing, 181

onetoone, 10

onto, 11

product of, 170

quotient of, 170

real, 170

scalarvalued, 170

sequences of, 227; see also Sequences of functions

series of, 228; see also Limits of series of functions

signum function (sgn), 156

strictly monotone, 182

sum of, 170

function value, 10

uniformly continuous, 197

vectorvalued, 170

Functions on E^{1}

antiderivatives of, 278

definite integrals of, 279

derivatives of, 251

derived, 251

differentials of, 288; see also Differentials of functions on E^{1}

differentiable, 251; see also Differentiable functions on E^{1}

exact primitives of, 278

of bounded variation, 303

indefinite integrals of, 278

integrable, 278; see also Integrable functions on E^{1}

length of, 301

Lipschitz condition for, 258

negative variation functions for, 308

nowhere differentiable, 253

positive variation functions for, 308

primitives of, 278

regulated, see Regulated functions

simple step, 323

step, 323

total variation of, 301

weakly absolutely continuous, 309

Functions on metric spaces,149

bounded, 111

continuity of composite, 163

continuity of the sum, product, and quotient of, 170

continuous, 149

discontinuous, 149

discontinuity points of, 149

domain space of, 149

limits of, 150

projection maps, 174, 198, 226

range space of, 149

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General term of a sequence, 16

Generator of an equivalence class, 13

Geometric series

limit of, 128, 236

sum of n terms of a, 33

Globes

closed globes in metric spaces, 97

deleted globes about points in metric spaces, 150

in E^{n}, 76

in E^{*}, 98

open globes in metric space, 97

Graphs of functions, 153

Greatest lower bound (glb) of a set in an ordered field, 38

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Halfclosed

interval in an ordered field, 37

interval in E^{n}, 77

line segment in E^{n}, 72

Halfopen

interval in an ordered field, 37

interval in E^{n}, 77

line segment in E^{n}, 72

Harmonic series, 241

Hausdorff property, 102

HeineBorel theorem, 324

generalized, 193

Holder's inequality, 93

Hyperharmonic series, 245, 329

Hyperplanes in E^{n}, 72. See also Planes in E^{n}

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iff (``if and only if''), 1

Image

of a set under a relation, 9

Imaginary

part of complex numbers, 81

numbers in C, 81

unit in C, 81

Inclusion relation of sets, 1

Increments

finite increments law, 271

of a function, 254

Independent

vectors in E^{n}, 70

Indeterminate expressions in E^{*}, 178

Index notation, 16. See also Sequence

Induction, 27

first induction law, 28

inductive definitions, 31; see also Inductive definitions

proof by, 29

second induction law, 30

Inductive definitions, 31

factorial, 31

powers with natural exponents, 31

ordered ntuple, 32

products of n field elements, 32

sum of n field elements, 32

Inductive sets in a field, 28

Infimum (inf) of a set in an ordered field, 38

Infinite

countably, 21

intervals, 54

sequence, 15

set, 18

Infinity

plus and minus, 53

unsigned, 179

Inner products
of vectors in E^{n}, 64

commutativity of, 67

distributive law of, 67

Integers in a field, 34

closure of addition and multiplication of, 35

Integrability, sufficient conditions for, 322. See also Regulated functions on intervals in E^{1}

Integrable functions on E^{1}, 278. See also Regulated functions on intervals in E^{1}

Dirichlet function, 329

primitively, 278

Integral part of elements of Archimedean fields, 44

Integral test of convergence of series, 315

Integrals

convergence of, 315

definite, 279; see also Definite integrals

indefinite, 278

Integration, 278

componentwise, 282

by parts, 281

of power series, 319

Interior

of a set in a metric space, 101

points of a set in a metric space, 101

Intermediate value property, 203

Intersection

of a family of sets, 3

of closed sets in metric spaces, 104

of open sets in metric spaces, 103

of sets, 2

Intervals in E^{n}, 76

boundary of, 77

center of, 77

closed, 77

degenerate, 78

diagonal of, 77

edgelengths of, 77

endpoints of, 77

halfclosed, 77

halfopen, 77

midpoints of, 77

open, 77

principle of nested, 189

volume of, 77

Intervals in E^{1}

partitions of, 300

Intervals in E^{*}, 54

finite, 54

infinite, 54


Intervals in an ordered field, 37

closed, 37

halfclosed, 37

halfopen, 37

open, 37

principle of nested, 42

Inverse elements

in a field, 24

of vector addition in E^{n}, 64, 65

Inverse function, see Inverse of a relation

continuity of the, 195, 207

derivative of the, 263

Inverse
image of a set under a relation, 9

Inverse pair, 8

Inverse of a relation, 8

Irrationals

density of irrationals in a complete field, 51

existence of irrationals in a complete field, 46

in a field, 34

Isometric metric spaces, 146

Isometry, 201. See also Functions

Iterated limits of functions, 221, 221

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Jumps of regulated functions, 330

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Kuratowski's definition of ordered pairs, 7

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Lagrange form of the remainder term of Taylor expansions, 291

Lagrange identity, 71

Lagrange's law of the mean, 262

Laws of the mean

Cauchy's, 261

first, 285

Lagrange's, 262

second, 286, 326

weighted, 286, 326

Leading term of a polynomial, 173

Least upper bound (lub) of a set in an ordered field, 37

Lebesgue number of a covering, 192

Left

bounded sets in an ordered field, 36

continuous functions, 153

derivatives of functions, 252

jump of a function, 184

limits of functions, 153

Leibniz

formula for derivatives of a product, 256

test for convergence of alternating series, 248

Length

function, 308

of arcs, 301, 311

of curves, 300

of functions, 301

of line segments in E^{n}, 72

of polygons, 300

of vectors in E^{n}, 64

L'Hospital's rule, 266

Limits of functions

Cauchy criterion for, 162

componentwise, 172

double, 219, 221

iterated, 221, 221

jointly, 174

left, 153

on E^{*}, 151

in metric spaces, 150

limits in one variable, 174

L'Hospital's rule, 266

relative, 152

relative, over a line, 174

right, 153

subuniform, 225

uniform, 220, 230

Limits of sequences

in E^{1}, 5, 54

in E^{*}, 55, 58, 152

in metric spaces, 115

lower, 56

subsequential limits, 135

upper, 56

Limits of sequences of functions

pointwise, 228

uniform, 228

Limits of series of functions

pointwise, 228

uniform, 228

Weierstrass Mtest, 240

Linear combinations of vectors in E^{n}, 66

Line segments in E^{n}, 72

closed, 72

endpoints of, 72

halfclosed, 72

halfopen, 72

length of, 72

midpoint of, 72

open, 72

principle of nested, 205

Linear functionals on E^{n}, 74, 75

equivalence between planes and nonzero, 76

representation theorem for, 75

Linear polynomials, 173

Linear spaces, see Vector spaces

Linearity of the definite integral, 280

Lines in E^{n}, 71

coordinate equations of, 72

directed, 74

direction vectors of, 71

normalized equation of, 73

parallel, 74

parametric equations of, 72

perpendicular, 74

symmetric form of the normal equations of, 74

Lipschitz condition, 258

Local

maximum and minimum of functions, 260

Logarithmic function, 208

continuity of the, 208

derivative of the, 263

integral definition of the, 331

as the inverse of the exponential function, 208

natural logarithm, 208

properties of the, 332

Logical formula, negation of a, 5

Logical quantifier, see Quantifier, logical

Lower bound of a set in an ordered field, 36

Lower limit of a sequence, 56

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Maclaurin series, 294

Mapping, see Function

contraction, 198

projection, 174, 198, 226

Master set, 2

Maximum

local, of a function, 260, 294

of a set in an ordered field, 36

Mean, laws of. See Laws of the mean

Metrics, 95. See also Metric spaces

axioms of, 95

discrete, 96

equivalent, 219

for E^{*}, 99

standard metric in E^{n}, 96

Metric spaces, 95. See also Metrics

accumulation points of sets or sequences in, 115

boundaries of sets in, 108

bounded functions on sets in, 111

bounded sequences in, 111

bounded sets in, 109

Cauchy sequences in, 141

Cauchy's convergence criterion for sequences in, 143

clopen sets in, 103

closed balls in, 97

closed sets in, 103, 138

closures of sets in, 137

compact sets in, 186

complete, 143

completion of, 146

concurrent sequences in, 144

connected, 212

constant sequences in, 116

continuity of the metric on, 223

convergent sequences in, 115

cluster points of sets or sequences in, 115

deleted globes about points in, 150

diameter of sets in, 109

disconnected, 212

dense subsets in, 139

discrete, 96

distance between sets in, 110

divergent sequences in, 115

E^{n} as a metric space, 96

E^{*} as a metric space, 98

functions on, 149; see also Functions on metric spaces

Hausdorff property in, 102

interior of a set in a, 101

interior points of sets in, 101

isometric, 146

limits of sequences in, 115

nowhere dense sets in, 141

open balls in, 97

open sets in, 101

open globes in, 97

neighborhoods of points in, 101

perfect sets in, 118

product of, 218

sequentially compact sets in, 186

spheres in, 97

totally bounded sets in, 113

Midpoints

of line segments in E^{n}, 72

of intervals in E^{n}, 77

Minimum

local, of a function, 260, 294

of a set in an ordered field, 36

Minkowski inequality, 94

Monomials in n variables, 173. See also Polynomials in n variables

degree of, 173

Monotone sequence of numbers, 17

nondecreasing, 17

nonincreasing, 17

strictly, 17

Monotone functions, 181

left and right limits of, 182

nondecreasing, 181

nonincreasing, 181

strictly, 182

Monotone sequence of sets, 17

Monotonicity

in an ordered field, 24

of definite integrals, 284

MooreSmith theorem, 223

de Morgan's duality laws, 3. See also Sets

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Natural elements in a field, 28

wellordering of naturals in an ordered field, 30

Natural numbers in E^{1}, 28

Negation of a logical formula, 5

Negative

elements of an ordered field, 25

variation functions, 308

Neighborhood

of a point in E^{1}, 58

of a point in E^{*}, 58

of a point in a metric space, 101

Neutral elements

in a field, 23

of vector addition in E^{n}, 65

Nondecreasing

functions, 181

sequences of numbers, 17

Nonincreasing

functions, 181

sequences of numbers, 17

Normal to a plane in E^{n}, 73

Normalized equations

of a line, 73

of a plane, 73

Normed linear spaces, 90

absolute value in, 90

C^{n} as a, 91

distances in, 92

E^{n} as a, 91

Euclidean spaces as, 91

norm in, 90

translationinvariant distances in, 93

triangle inequality in, 90

Norms

in normed linear spaces, 90

standard norm in C^{n}, 91

standard norm in E^{n}, 91

Nowhere dense sets in metric spaces, 141

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Open

ball in a metric space, 97

covering, 192

globe in a metric space, 97

interval in an ordered field, 37

interval in E^{n}, 77

line segment in E^{n}, 72

sets in a metric space, 101

Ordered field, see Field, ordered

Ordered ntuple, 1

inductive definition of an, 32

Ordered pair, 1

inverse, 8

Kuratowski's definition of an, 7

Orthogonal vectors in E^{n}, 65

Orthogonal projection

of a point onto a plane in E^{n}, 76

Osgood's theorem, 221, 223

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Parallel

lines in E^{n}, 74

planes in E^{n}, 74

vectors in E^{n}, 65

Parametric equations

of curves in E^{n}, 212

of lines in E^{n}, 72

Partitions of intervals in E^{1}, 300

refinements of, 300

Pascal's law, 34

Peano form of the remainder term of Taylor expansions, 296

Perfect sets in metric spaces, 118

Cantor's set, 120

Perpendicular

lines in E^{n}, 74

planes in E^{n}, 74

vectors in E^{n}, 65

Piecewise convex sets, 204

Planes in E^{n}, 72

directed, 74

distance between points and, 76

equation of, 73

equivalence of nonzero linear functionals and, 76

general equation of, 73

normal to, 73

normalized equations of, 73

orthogonal projection of a point onto, 76

parallel, 74

perpendicular, 74

Point in E^{n}, 63

distance from a plane to a, 76

orthogonal projection onto a plane, 76

Pointwise limits

of sequences of functions, 228

of series of functions, 228

Polar coordinates in C, 83

Polar form of complex numbers, 83

Polygons

connected sets, 204

joining two points, 204

length of, 300

Polygonconnected sets, 204

Polynomials in n variables, 173

continuity of, 173

degree of, 173

leading term of, 173

linear, 173

Positive

elements of an ordered field, 25

variation functions, 308

Power function, 208

continuity of the, 209

derivative of the, 264

Power series, 243

Abel's theorem for, 249

differentiation of, 319

integration of, 319

radius of convergence of, 243

Taylor series, 292

Powers

with natural exponents in a field, 31

with rational exponents in a complete field, 47

with real exponents in a complete field, 50

Primitive, 278. See also Integral, indefinite

exact, 278

Principle of nested

closed sets, 188

intervals in complete ordered fields, 189

intervals in E^{n}, 189

intervals in ordered fields, 42

line segments, 205

Products of functions, 170

derivatives of, 256

Leibniz formula for derivatives of, 256

Product of metric spaces, 218

Projection maps, 174, 198, 226

Proper subset of a set, 1

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Quantifier, logical, 3

existential, 4

universal, 4

Quotient of elements of a field, 26

Quotient of functions, 170

derivatives of, 256

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Radius of convergence of a power series, 243

Range

of a relation, 9

of a sequence, 16

space of functions on metric spaces, 149

Ratio test for convergence of series, 241

Rational functions, 173

continuity of, 173

Rational numbers, 19

as a countable set, 19

Rationals

closure laws of, 35

density of rationals in an Archimedean field, 45

incompleteness of, 47

in a field, 34

as a subfield, 35

Real

functions, 170

numbers, see E^{1}

part of complex numbers, 81

points in C, 81

vector spaces, 87

unit in C, 81

Rearrangement

of absolutely convergent series of functions, 238

of conditionally convergent series of functions, 250

Rectifiable

arc, 309

set, 303

Recursive definition, 31. See also
Inductive definition

Refined comparison test for convergence of series, 245

Refinements of partitions in E^{1}, 300

Reflexive relation, 12

Regulated functions on intervals in E^{1}, 323

approximation by simple step functions, 324

characteristic functions of intervals, 323

jumps of, 330

are integrable, 325

simple step functions, 323

Relation, 8. See also Sets

domain of a, 9

equivalence, 12

image of a set under a, 9

inverse, 8

inverse image of a set under a, 9

range of a, 9

reflexive, 12

symmetric, 12

transitive, 12

Relative

continuity of functions, 152, 174

limits of functions, 152, 174

Remainder term of Taylor expansions, 289

Cauchy form of the, 291

integral form of the, 289

Lagrange form of the, 291

Peano form of the, 296

SchloemilchRoche form of the, 296

Representative of an equivalence class, 13

Right

bounded sets in an ordered field, 36

continuous functions, 153

derivatives of functions, 252

jump of a function, 184

limits of functions, 153

Rolle's theorem, 261

Root test for convergence of series, 241

Roots

in C, 85

in a complete field, 46

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Scalar field of a vector space, 86

Scalar products

in E^{n}, 64

Scalarvalued functions, 170

Scalars

of E^{n}, 64

of a vector space, 86

SchloemilchRoche form of the remainder term of Taylor expansions, 296

Second induction law, 30

Second law of the mean, 286, 326

Sequences, 15

bounded, 111

Cauchy, 141

Cauchy's convergence criterion for, 143

concurrent, 144

constant, 116

convergent, 115

divergent, 115

domain of, 15

double, 20, 222, 223

cluster points of sequences in E^{*}, 60

finite, 16

general terms of, 16

index notation, 16

infinite, 15

limits of sequences in E^{1}, 5, 54

limits of sequences in E^{*}, 55, 58, 152

limits of sequences in metric spaces, 115

lower limits of, 56

monotone sequences of numbers, 17

monotone sequences of sets, 17

nondecreasing sequences of numbers, 17

nonincreasing sequences of numbers, 17

range of, 16

of functions, 227; see also Sequences of functions

strictly monotone sequences of numbers, 17

subsequences of, 17

subsequential limits of, 135

totally bounded, 188

upper limits of, 56

Sequences of functions

limits of, see Limits of sequences of functions

uniformly bounded, 234

Sequential criterion

for continuity, 161

for uniform continuity, 203

Sequentially compact sets, 186

Series. See also Series of functions

Abel's test for convergence of, 247

alternating, 248

geometric, 128, 236

harmonic, 241

hyperharmonic, 245, 329

integral test of convergence of, 327

Leibniz test for convergence of alternating series, 248

ratio test for convergence of, 241

refined comparison test, 245

root test for convergence of, 241

summation by parts, 247

Series of functions, 228; see also Limits of series of functions

absolutely convergent, 237

conditionally convergent, 237

convergent, 228

Dirichlet test, 248

differentiation of, 318

divergent, 229

integration of, 318

limit of geometric series, 128

power series, 243; see also Power series

rearrangement of, 238

sum of n terms of a geometric series, 33

Sets, 1

Cantor's diagonal process, 21

Cantor's set, 120

Cartesian product of, 2

characteristic functions of, 323

compact, 186, 193

complement of a set, 2

connected, 212

convex, 204

countable, 18

countable union of, 20

cross product of, 2

diagonal process, Cantor's, 21

difference of, 2

disjoint, 2

distributive laws of, 7

contracting sequence of, 17

elements of, 1

empty set, 1

equality of, 1

expanding sequence of, 17

family of, 3

finite, 18

inclusion relation of, 1

infinite, 18

intersection of a family of, 3

intersection of, 2

master set, 2

monotone sequence of, 17

de Morgan's duality laws, 3

perfect sets in metric spaces, 118

piecewise convex, 204

polygonconnected, 204

proper subset of a set, 1

rectifiable, 303

relation, 8

sequentially compact, 186

subset of a set, 1

superset of a set), 1

uncountable, 18

union of a family of, 3

union of, 2

Signum function (sgn), 156

Simple arcs, 211

endpoints of, 211

Simple step functions, 323

approximating regulated functions, 324

Singleton, 103

Span of a set of vectors in a vector space, 90

Sphere

in E^{n}, 76

in a metric space, 97

Step functions, 323

simple, 323

Strictly monotone functions, 182

Subsequence of a sequence, 17

Subsequential limits, 135

Subset of a set, 1

proper, 1

Subuniform limits of functions, 225

Sum of functions, 170

Summation by parts, 247

Superset of a set, 1

Supremum (sup) of a bounded set in an ordered field, 38

Symmetric relation, 12

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Tangent

lines to curves, 257

vectors to curves, 257

unit tangent vectors, 314

Taylor. See also Taylor expansions

expansions, 289

polynomial, 289

series, 292; see also power series

series about zero (Maclaurin series), 294

Taylor expansions, 289. See also Remainder term of Taylor expansions

for the cosine function, 297

for the exponential function, 293

for the logarithmic function, 298

for the power function, 298

for the sine function, 297

Termwise

differentiation of series of functions, 318

integration of series of functions, 318

Total variation, 301

additivity of, 301

function, 308

Totally bounded sets in metric spaces, 113

Totally disconnected sets, 217

Transitive relation, 12

Transitivity in an ordered field, 24

Triangle inequality

in Euclidean spaces, 88

in normed linear spaces, 90

of the absolute value in E^{n}, 67

of the distance in E^{n}, 68

Trichotomy in an ordered field, 24

Trigonometric form of complex numbers, 83

Trigonometric functions

arcsine, 334

cosine, 336

integral definitions of, 334

sine, 336

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Uncountable set, 18

Cantor's diagonal process, 21

the real numbers as a, 20

Uniform continuity, 197

sequential criterion for, 203

Uniform limits

of functions, 220, 230

of sequences of functions, 228

of series of functions, 228

Uniformly continuous functions, 197

Union

countable, 20

of a family of sets, 3

of closed sets in metric spaces, 104

of open sets in metric spaces, 103

of sets, 2

Unit vector

tangent, 314

in E^{n}, 65

Universal quantifier, 4

Unorthodox operations in E^{*}, 180

Upper bound of a set in an ordered field, 36

Upper limit of a sequence, 56

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Variation

bounded, 303

negative variation functions, 308

positive variation functions, 308

total; see Total variation

Vectorvalued functions, 170

Vectors in E^{n}, 63

absolute value of, 64

angle between, 70

basic unit, 64

components of, 63

coordinates of, 63

dependent, 69

difference of, 64

distance between two, 64

dot product of two, 64

independent, 70

inner product of two, 64; see also
Inner products of vectors in E^{n}

inverse of, 65

length of, 64

linear combination of, 66

orthogonal, 65

parallel, 65

perpendicular, 65

sum of, 64

unit, 65

zero, 63

Vector spaces, 86

complex, 87

Euclidean spaces, 88

normed linear spaces, 90

real, 87

scalar field of, 86

span of a set of vectors in, 90

Volume of an interval in E^{n}, 77

additivity of the, 79

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Weierstrass Mtest for convergence of series, 240

Weighted law of the mean, 286, 326

Wellordering property, 30

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Zero vector in E^{n}, 63
