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Index of
Mathematical Analysis I
by Elias Zakon

To indicate the range of topics covered in the award-winning electronic text Mathematical Analysis I by Elias Zakon, we include here the book's index.

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z
Abel's convergence test, 247
Abel's theorem for power series, 249, 322
Absolute value
in an ordered field, 26
in En, 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 En, 79
Alternating series, 248
Admissible change of variable, 165
Angle between vectors in En, 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
Arithmetic-geometric mean, Gauss's, 134
Associative laws
in a field, 23
of vector addition in En, 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 En, 64
Bernoulli inequalities, 33
Binary operations, 12. See also Functions
Binomial theorem, 34
Bolzano theorem, 205
Bolzano-Weierstrass theorem, 136
Boundary
of intervals in En, 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
left-bounded sets in ordered fields, 36
right-bounded 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
Cn (complex n-space), 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 En as Cartesian products of intervals in E1, 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
Cauchy-Schwarz inequality
in En, 67
in Euclidean spaces, 88
Center of an interval in En, 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 En, 77
line segment in En, 72
sets in metric spaces, 103, 138
Closures of sets in metric spaces, 137
Closure laws
in a field, 23
in En, 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 En, 65
of inner products of vectors in En, 67
Compact sets, 186, 193
Cantor's principle of nested closed sets, 188
are totally bounded, 188
in E1, 195
continuity on, 194
generalized Heine-Borel theorem, 193
Heine-Borel 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 E1, 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 E1, 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 M-test 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 En, 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 En, 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 En, 69
Derivatives of functions on E1, 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 extended-real values, 259
left, 252
one-sided, 252
right, 252
Derived functions on E1, 251
nth, 252
Diagonal of an interval in En, 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 E1, 288
of order n, 289
Differentiable functions on E1, 251
Cauchy's laws of the mean, 261
cosine function, 337
are continuous, 252
exponential function, 333
infinitely, 292
logarithmic function, 332
n-times continuously, 292
n-times, 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 En, 74
planes in En, 74
Direction vectors of lines in En, 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 En, 76
between sets in metric spaces, 110
between two vectors in En, 64
between two vectors in Euclidean spaces, 89
in normed linear spaces, 92
norm-induced, 92
translation-invariant, 93
Distributive laws
in En, 65
in a field, 24
of inner products of vectors in En, 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 Cn, 87
in En, 64
Duality laws, de Morgan's, 3. See also Sets

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e (the number), 122, 165, 293
E1 (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
En (Euclidean n-space), 63. See also Vectors in En
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
Bolzano-Weierstrass theorem, 136
Cauchy-Schwarz 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 En
intervals in, 76; see also Intervals in En
line segments in, 72; see also Line segments in En
linear functionals on, 74, 75; see also Linear functionals on En
lines in, 71; see also Lines in En
neutral element of vector addition in, 65
planes in, 72; see also Planes in En
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
Edge-lengths of an interval in En, 77
Elements of a set, 1
Empty set, 1
Endpoints
of an interval in En, 77
of line segments in En, 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 n-space, see En
Euclidean spaces, 88
as normed linear spaces, 91
absolute value in, 88
Cn as a, 88
Cauchy-Schwarz inequality in, 88
distance in, 89
En 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
half-closed intervals in, 37
half-open 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
well-ordering 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 E1 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
one-to-one, 10
onto, 11
product of, 170
quotient of, 170
real, 170
scalar-valued, 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
vector-valued, 170
Functions on E1
antiderivatives of, 278
definite integrals of, 279
derivatives of, 251
derived, 251
differentials of, 288; see also Differentials of functions on E1
differentiable, 251; see also Differentiable functions on E1
exact primitives of, 278
of bounded variation, 303
indefinite integrals of, 278
integrable, 278; see also Integrable functions on E1
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 En, 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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Half-closed
interval in an ordered field, 37
interval in En, 77
line segment in En, 72
Half-open
interval in an ordered field, 37
interval in En, 77
line segment in En, 72
Harmonic series, 241
Hausdorff property, 102
Heine-Borel theorem, 324
generalized, 193
Holder's inequality, 93
Hyperharmonic series, 245, 329
Hyperplanes in En, 72. See also Planes in En

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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 En, 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 n-tuple, 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 En, 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 E1
Integrable functions on E1, 278. See also Regulated functions on intervals in E1
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 En, 76
boundary of, 77
center of, 77
closed, 77
degenerate, 78
diagonal of, 77
edge-lengths of, 77
endpoints of, 77
half-closed, 77
half-open, 77
midpoints of, 77
open, 77
principle of nested, 189
volume of, 77
Intervals in E1
partitions of, 300
Intervals in E*, 54
finite, 54
infinite, 54
Intervals in an ordered field, 37
closed, 37
half-closed, 37
half-open, 37
open, 37
principle of nested, 42
Inverse elements
in a field, 24
of vector addition in En, 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 En, 72
of polygons, 300
of vectors in En, 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 E1, 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 M-test, 240
Linear combinations of vectors in En, 66
Line segments in En, 72
closed, 72
endpoints of, 72
half-closed, 72
half-open, 72
length of, 72
midpoint of, 72
open, 72
principle of nested, 205
Linear functionals on En, 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 En, 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 En, 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
En 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 En, 72
of intervals in En, 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
Moore-Smith theorem, 223
de Morgan's duality laws, 3. See also Sets

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Natural elements in a field, 28
well-ordering of naturals in an ordered field, 30
Natural numbers in E1, 28
Negation of a logical formula, 5
Negative
elements of an ordered field, 25
variation functions, 308
Neighborhood
of a point in E1, 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 En, 65
Nondecreasing
functions, 181
sequences of numbers, 17
Nonincreasing
functions, 181
sequences of numbers, 17
Normal to a plane in En, 73
Normalized equations
of a line, 73
of a plane, 73
Normed linear spaces, 90
absolute value in, 90
Cn as a, 91
distances in, 92
En as a, 91
Euclidean spaces as, 91
norm in, 90
translation-invariant distances in, 93
triangle inequality in, 90
Norms
in normed linear spaces, 90
standard norm in Cn, 91
standard norm in En, 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 En, 77
line segment in En, 72
sets in a metric space, 101
Ordered field, see Field, ordered
Ordered n-tuple, 1
inductive definition of an, 32
Ordered pair, 1
inverse, 8
Kuratowski's definition of an, 7
Orthogonal vectors in En, 65
Orthogonal projection
of a point onto a plane in En, 76
Osgood's theorem, 221, 223

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Parallel
lines in En, 74
planes in En, 74
vectors in En, 65
Parametric equations
of curves in En, 212
of lines in En, 72
Partitions of intervals in E1, 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 En, 74
planes in En, 74
vectors in En, 65
Piecewise convex sets, 204
Planes in En, 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 En, 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
Polygon-connected 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 En, 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 E1
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 E1, 300
Reflexive relation, 12
Regulated functions on intervals in E1, 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
Schloemilch-Roche 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 En, 64
Scalar-valued functions, 170
Scalars
of En, 64
of a vector space, 86
Schloemilch-Roche 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 E1, 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
polygon-connected, 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 En, 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 En, 67
of the distance in En, 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 En, 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
Vector-valued functions, 170
Vectors in En, 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 En
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 En, 77
additivity of the, 79

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Weierstrass M-test for convergence of series, 240
Weighted law of the mean, 286, 326
Well-ordering property, 30

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Zero vector in En, 63
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