one-sided, 203
vs. one-sided limit, 206, 207
limit of a sequence, 52, 61, 64
lower (inferior), 132 , 133
upper (superior), 131, 133
limits preserve inequalities
for functions, 198, 206
for functions at infinity, 220
for sequences, 79
lim sup of a sequence, 133
linearity of sums of series, 457
linearity of the integral, 397
Lipschitz condition, 265, 305, 329
local extreme value of a function,
316 , 318
local maximum or minimum of a
function, 316, 318
locally bounded function, 164, 235
location of roots principle, 252
logarithm functions, 288, 289, 422 ,
427, 428, 538
defined as integral, 422
derivative of, 312, 313, 422
properties of, 288, 423
logical connectives, 584
lower bound, 35
greatest, 37
lower Darboux integral, 363
lower Darboux sum, 361
lower limit of a sequence, 132m-tail of a sequence, 69
Maclaurin series, 512
mathematical induction, 22
alternate principle of, 24
as a method of definition, 26
for n ~ n 0 , 25
maximum, 19, 35
of two functions, 626
meager set, 294
mean
arithmetic, 19 , 72
arithmetic-geometric, 116
geometric, 20, 73
harmonic, 20
quadratic, 20Index 733recursive arithmetic, 116, 120
recursive geometric, 116
recursive weighted arithmetic,
121
mean value theorem, 323, 334
mean value theorem for integrals
first, 413
second, 413
measure (Lebesgue), 171
measure zero, 170, 444
Mertens' Theorem, 491
mesh of a partition, 372
minimum, 19 , 35
of two functions, 627
monotone convergence theorem, 91
monotone increasing/ decreasing
function, 240, 318 , 325, 368, 393
sequence, 89
multiplicative inverse, 6N, 4, 24
n-tuple or n-vector, 498
natural numbers, 4, 24
of an ordered field, 21
negation, 588
of an implication, 593
of limit statement, 183
of quantifiers, 600
of uniform continuity, 260
negative element, 11
neighborhood, 138
deleted, 188
of infinity, 219
neighborhood inequality property
of continuous functions, 235
of limits, 202
Np, 21
nonarchimedean ordered field, 34
nonintegrable function, 364, 398
norm of function, 548
nowhere dense set, 169 , 175, 293
nth derivative test, 338, 340