INDEX 391

measuresdecreasing sequences of sets, 324increasing sequences of sets, 324

Merten’s theorem, 98, 185modular group, 273monic

polynomial, 35monotone, 216monotone convergence theorem, 339monotone decreasing, 75monotone function

differentiable, 349monotone increasing, 75Morera’s theorem, 256multinomial expansion, 30

nested closed setsshrinking diameters, 80

nested interval lemma, 65nonremovable discontinuity, 101norms, 377nowhere differentiable function, 139

odd, 293open set, 68

connected components, 112order, 13ordered

partial, 50totally ordered, 50

oriented interval, 190outer measure, 346

p series, 90p variation, 376

approximation, 379partial fractions

expansion, 40theory, 40

partial order, 50partial summation formula, 92partially ordered set, 50partition

ordered, 252perfect sets, 114periodic function, 279permutations, 18permutations and combinations, 30pi systems, 326

piecewise continuous, 192, 284piecewise smooth curve, 251pointwise convergence

sequence, 116series, 119

pole, 260, 264polynomial, 34

addition, 34degree, 34division, 34equality, 34leading term, 34monic, 34, 35multiplication, 34root, 34zero, 34

polynomialsgreatest common divisor, 35relatively prime, 35

power series, 157exponential sine and cosine, 162multiplication, 179of a quotient, 188raising to a power, 186

power set, 43, 52present value of an annuity, 25preservation of compactness, 108prime number, 22

less than a given number, 100prime numbers

infinitely many, 26product rule, 139

quadratic formula, 31quotient rule, 141

Raabe’s test, 153Rademacher’s theorem, 353radius of convergence, 158raising a number to a power, 167range, 55ratio test, 94rational function, 57rational numbers, 9

density, 21real numbers, 9

construction from Rationals, 357rearranged series