414 INDEX

density, 21real numbers, 9

construction from Rationals, 385rearranged series

convergence, 90recurrence relation, 58recursively defined sequence, 83recursively defined sequence, 58region, 277regular measures, 341regular Sturm Liouville problem, 229relatively closed, 109relatively open, 109relatively prime, 22removable discontinuity, 105removable singularity, 287residue, 272residues

method, 272Rieman Stieltjes integral

existence, 206Riemann integral

improper, 235without using sums, 159

Riemann Lebesgue lemma, 240Riemann Stieltjes integral

existence, 201properties, 200uniqueness, 200

Rolle’s theorem, 150root test, 97, 98roots

existence, 28roots of complex number, 188Russell’s paradox, 45

Schroder Bernstein theorem, 47second derivative test, 158second mean value theorem, 230

integrals, 243separated, 114sequence

functions, 120sequence of partial sums, 87sequences, 57

terminating, 53uncountable, 52

sequential compactness, 118

sequentially compactclosed and bounded, 71in complex numbers, 71

seriesabsolute convergence, 89conditional convergence, 89convergence criterion, 89double sum, 99meaning of convergence, 87multiplication of series, 101nonnegative terms, 87p series test, 93ratio test, 97root test, 98

series of functionsuniform convergence, 123

sets, 43sigma algebra, 333simple function, 343Simpson’s rule, 227, 228smooth and not analytic, 183smooth curve, 257smoothness not analytic, 182space filling curve, 129squeezing theorem, 134squeezing theorem, 63star shaped, 265step function, 197Stirling’s formula, 219, 368

real arguments, 255sup

increasing sequence, 100interchange of order, 28

symmetric polynomial theorem, 394symmetric polynomials, 393

tagged division, 313Tarski, 53Taylor series, 161, 267

coefficients, 165convergence and divergence, 161differentiation, 163exponential sine and cosine, 166multiplication, 183of quotient, 192raising to a power, 190uniqueness, 165

Taylor’s formula, 152, 228