Elementary Real Analysis, 2Nd Edition (2008) Subject Index Classicalrealanalysis.Com A-29

ClassicalRealAnalysis.com

SUBJECT INDEX

Rn neighborhood, 652 ε-neighborhood, 652 open ball, 652 accumulation point, 652 open cover, 680 Bolzano-Weierstrass theorem, 659 open set, 652 boundary point, 653 orthogonal, 646 bounded sequence, 656 sequence, 656 bounded set, 653 uniform continuity, 681 Cauchy criterion, 661 Rn closed set, 652 linear function, 663 closure of a set, 654 compact set, 680 Abel connected set, 655, 684 summability method, 193 convergent sequence, 656 test for uniform convergence, 555 dot product, 646 Abel, N., 108 euclidean distance, 647 absolute convergence, 136 euclidean norm, 647 of power series, 598 Heine Borel property, 680 absolute value, 22 limit of a function, 668 absolutely closed, 858 limit point, 652 absolutely convergent integral, 505, 508, 529 accumulation point, 222

A-28 Thomson*Bruckner*Bruckner Elementary Real Analysis, 2nd Edition (2008) Subject Index ClassicalRealAnalysis.com A-29

in Rn, 652 binomial series, 161 algebra of functions, 795 Bolzano, B., 80 algebraic number, 41 Bolzano-Weierstrass property, 241 algebraic properties of sequence limits, 53 Bolzano-Weierstrass theorem alternating harmonic series, 125 in Rn, 659 analytic function, 482, 620 in R, 80 annuity, 111 in a metric space, 883, 889 antiderivative, 485 Borel set, 372 archimedean property, 16 Borel, E., 248, 372 arithmetic progression, 33 boundary point, 223 arithmetic-geometric mean inequality, 10 in Rn, 653 Arzelà,C., 897, 901 of a set in a metric space, 817 Arzelà-Ascolitheorem, 897, 901 bounded Ascoli, G., 896 function, 15 associated points, 489 sequence, 49 axiom of completeness, 13 sequence in Rn, 656 axioms set in Rn, 653 for a field, 6 set in a metric space, 817 for a partial order, 796 set of real numbers, 11 totally bounded, 892 Baire category theorem, 359, 911 uniformly bounded, 896 Baire, R., 356 bounded variation (sequences), 138 Baire 1 function, 589 boundedness criterion for series, 132 Banach fixed point theorem, 861 boundedness property for limits, 280 Banach, S., 356, 921 Brouwer’s fixed point theorem, 859 Banach-Mazur game, 356 Brouwer, L., 859 Bernstein’s theorem, 623 Bessel function, 615 Cantor Bessel, F. W., 932 characteristic function of Cantor set, 300 bijective function, A-7 complementary intervals of Cantor set, 365 binary operation, 8 function, 369, 370

Thomson*Bruckner*Bruckner Elementary Real Analysis, 2nd Edition (2008) A-30 ClassicalRealAnalysis.com Subject Index

intersection property, 243, 854 sequence in a metric space, 850 set, 365, 366, 368, 370, 375, 376, 425, 921 Cauchy, A., 305 space, 841 Cauchy-Riemann equations, 694 ternary set, 239, 363, 365 Cauchy-Schwarz inequality, 138, 527, 534, 648, theorem, 38 804 Cantor, G., 2, 38, 217, 370 Cesàrosummability method, 190 cardinally equivalent, 40 Cesàro,E., 191 careless student, 59, 65, 87, 94, 233, 286, 369, 381, chain rule, 411, 729 391, 531, 533, 546, 678, 722, 737, 839, characteristic function, 272, 296, A-8 866 of Cantor set, 300 Cartesian product, A-5 Chebychev polynomials, 639 category Chebychev, P. L., 639 first, 359, 912 circular sums, 198 residual, 359, 912 class, see set second, 359, 912 closed Cauchy ball, 816 form, 106 belief in term by term integration, 549 interval, A-2 criterion for absolutely convergent integrals, relative to a set, 258 510 set, 227 criterion for convergent integrals, 509 set in Rn, 652 criterion for sequences, 84 set in a metric space, 817 criterion for series, 133 closed unit ball, 662 criterion for uniform convergence, 551 closure criterion for unordered sum, 115 of a set in Rn, 654 first method, 489 of a set in a metric space, 817 mean value theorem, 434 of a set of real numbers, 228 scope of first method, 492 cluster point, 83 second method, 504, 507 cluster value of a function, 342 sequence, 85 co-countable set, 257

Thomson*Bruckner*Bruckner Elementary Real Analysis, 2nd Edition (2008) Subject Index ClassicalRealAnalysis.com A-31 coefficients continuity formulas for Fourier coefficients, 630 and uniform convergence, 564 of power series, 594, 610 modulus of continuity, 900 collage theorem, 873 of a derivative, 421 collection, see set of differentiable function, 404 common difference of an arithmetic progression, of power series, 606 33 set of points of, 382 common ratio of a geometric progression, 34 uniform continuity, 887 compact set, 252 continuous in Rn, 680 a.e., 518 in a metric space, 883 almost everywhere, 518 compactness argument, 241 at a point, 309 comparison of series and sums, 177 convergence, 568 complete metric space, 851 equicontinuous family of functions, 896 completeness function, 294 axiom, 13 function from Rn to Rm, 676 proofs, 851 function has max and min, 326 completion of a metric space, 854 function in a metric space, 828 complex numbers, 25 on a set, 316 component intervals of open set, 230 separately continuous, 832 composition uniformly, 322 of functions, 292, 411, A-8 contraction of power series, 628 map, 859 concave function, 449 weak contraction, 888 concave-down, 449 contractive sequence, 97 concave-up, 449 contraposition, A-17 congruence, 843 convergence connected metric space, 841 continuous, 568 connected set, 260 coordinate-wise, 811 in Rn, 655, 684 dominated, 576

Thomson*Bruckner*Bruckner Elementary Real Analysis, 2nd Edition (2008) A-32 ClassicalRealAnalysis.com Subject Index

in a metric space, 811 crossing lines, 915 of a sequence, 41 cube, Hilbert, 813, 824 of alternating harmonic series, 125 curve, 833 pointwise, 540 pointwise convergence of a sequence, 539 Darboux uniform convergence, 549 function, 305 convergent integral, 536 integral, 504, 529 property, 328, 445 sequence, 656 property of derivative, 444 sequence is bounded, 49 sums, 536 series, 121 Darboux, J. G., 306, 444 subsequence, 80 De Morgan’s laws, A-4 unordered sum, 113 decimal expansions, 75 converges infinitely slowly, 548 decreasing converse, A-18 function, 333, 435 convex function, 449 sequence, 66 coordinate functions, 665 Dedekind, R., 2 coordinate-wise convergence, 811 definite integral, 485 countable set, 37, 39, 256 definition of for continuous functions, 489 counterexample, A-18, A-19 deleted neighborhood, 220 Cousin cover, 246 Denjoy, A., 473 Cousin’s lemma, 246 Denjoy-Young-Saks theorem, 473 Cousin, P., 245 dense cover in a metric space, 818 Cousin, 246 in a set, 26, 352 open cover, 248, 889 dense in itself, 825 covering theorems, 246 dense set, 20, 352 critical point, 424 dense-in-itself, 257 cross partials, 697 derivative, 398, 778 cross product, 684 and local extrema, 424 and monotonicity, 436

Thomson*Bruckner*Bruckner Elementary Real Analysis, 2nd Edition (2008) Subject Index ClassicalRealAnalysis.com A-33

and uniform convergence, 578 differentiable antiderivative, 485 continuity of differentiable function, 404 as a magnification, 405 function, 398, 714 chain rule, 411 function from Rn to Rm, 778 computation of, 407 differential, 726, 778 Darboux property of, 444 differential equation, 870, 878, 901 Dini derivates, 440 Dini derivates, 440 directional, 695 Dini’s theorem, 566 higher-order derivative, 401 Dini, U., 440, 566 left-hand derivative, 399 Dirichlet function, 297 not continuous, 421 Dirichlet kernel, 637 not the limit of derivatives, 541 disconnected set, 260 of inverse function, 417, 445 discontinuity of power series, 609 essential, 332 power rule, 418 jump, 276, 295, 331 product of, 447 of a limit of continuous functions, 541 right-hand derivative, 399 of monotonic functions, 334 second derivative, 401 removable, 331 straddled, 471 discrete metric, 799 symmetric derivative, 402 distance trapping principle, 400 between a point and a set, 299, 818 unstraddled, 471 between sets, 16, 259, 810 derived function, 22 number, 443 divergent set, 226 harmonic series, 125 descending sequence of sets, 243 integral, 504 devil’s staircase, 369 sequence, 42 diameter, 817 series, 121 difference of sets, A-3 to infinity, 47 difference quotient, 396 domain of a function, A-7

Thomson*Bruckner*Bruckner Elementary Real Analysis, 2nd Edition (2008) A-34 ClassicalRealAnalysis.com Subject Index

dominated convergence theorems, 576 product, 111 dot product, 646 sums, 105 dyadic rationals, 21 first category set, 359, 912 fixed point, 330, 859 elliptic integral, 708 four-color problem, 683 embedding, 845 Fourier coefficients, 630 empty set , A-2 ∅ Fourier series, 629, 632 enumeration, 37 uniform convergence of, 630 equicontinuty, 896 Fourier, J., 630 equivalence relation, 40, 838, A-12 Fréchet, M., 798 essential discontinuity, 332 Fredholm’s integral equation, 869 euclidean Fritsch, G., 683 distance, 647 Fritsch, R., 683 metric, 649 function norm, 647 analytic, 482, 620 plane, 799 Baire 1, 589 Euler’s constant, 215 Bessel function, 615 Euler, L., 105 bijective, A-7 even function, 615 bounded, 15 extension of a function, 341 Cantor function, 370 extreme unilateral derived numbers, 443 characteristic function of a set, A-8 characteristic of rationals, 272, 296 family, see set cluster value of, 342 Fejérkernel, 634 composition, 411, A-8 Fejér,L., 634 concave, 449 Fibonacci sequence, 37, 97 continuity of in a metric space, 828 field, 6 continuous, 294 ordered field, 10 continuous at a point, 309 field axioms, 6 continuous on a set, 316 finite contraction, 859 intersection property, 259

Thomson*Bruckner*Bruckner Elementary Real Analysis, 2nd Edition (2008) Subject Index ClassicalRealAnalysis.com A-35

convex, 449 left continuous, 319 crossing lines, 915 lim inf of, 302 decreasing, 333, 435 lim sup of, 302 decreasing function, 334 limit of, 264, 269, 272, 668 definition of, A-6 limit of a composition, 292 derivative of, 398 linear, 663 derived number of, 443 Lipschitz, 316, 431 difference quotient, 396 locally bounded, 240 differentiable, 398, 714, 778 mapping, 765, A-10 differential, 726 monotonic, 333, 435 Dirichlet, 297 nondecreasing, 333, 435 discontinuities of, 298 nondecreasing function, 333 distance between set and point, 299 nonincreasing, 333, 435 domain of, A-7 nonincreasing function, 334 equicontinuous family of functions, 896 nowhere differentiable, 921 even, 615 nowhere monotonic, 920 extension of, 341 odd, 615 fixed point of, 330, 859 of Cantor, 369 graph of, A-6 of Pompeiu, 585 greatest integer function, 14 one-to-one, A-7 harmonic, 693, 694, 745 onto, A-7 image, A-10 oscillation at a point, 379 implicit, 749 oscillation of, 378 increasing, 333, 435 oscillation on an interval, 378 increasing function, 333 pointwise bounded family, 361 infinite limit, 277 preimage, A-11 inflection point, 453 range of, A-7 injective, A-7 right continuous, 319 intermediate value property, 305 right-hand limit, 274 inverse of, 416, A-8 smooth, 433

Thomson*Bruckner*Bruckner Elementary Real Analysis, 2nd Edition (2008) A-36 ClassicalRealAnalysis.com Subject Index

step function, 298 higher-order derivative, 401 surjective, A-7 Hilbert cube, 813, 824 transformation, 765 Hilbert space, 686, 803 uniformly bounded family, 361 Hilbert, D., 803 uniformly continuous, 322, 681, 887 homeomorphic, 836 uniqueness of limit, 279 homeomorphism, 366, 836 vector-valued, 664 Hubbard, J. H., 683 function space, 806 functions I. P. Natanson, I., 589 series, 538 image, A-10 uniformly bounded, 361 implicit function, 749 fundamental theorem of calculus, 499, 525 implies , A-17 improper⇒ Riemann integral, 528 generalized Riemann integral, 532 increasing geometric progression, 34, 72 function, 333, 435 formula for sum, 108 sequence, 66 geometric series, 124 indefinite integral, 485 Gordon, R. A., 532 indicator function, see characteristic function of a gradient, 718 set greatest integer function, 14 indirect proof, A-15 greatest lower bound, 13 induction, 19, A-19 inductive set, 26 harmonic function, 693, 694, 745 inequality harmonic series, 125 arithmetic-geometric mean inequality, 10 Hausdorff metric, 810 Cauchy-Schwarz, 138, 527, 534, 804 Hausdorff, F., 798 triangle inequality, 24 Heine, E., 217, 248 inf, see infimum Heine-Borel property, 248, 889 infimum, 13 in Rn, 680 infinite Heine-Borel theorem, 248 decimals, 5 Henstock, R., 532 limit, 277

Thomson*Bruckner*Bruckner Elementary Real Analysis, 2nd Edition (2008) Subject Index ClassicalRealAnalysis.com A-37

product, 201 of Darboux, 536 sum (unordered), 112 of Lebesgue, 512 infinitely slowly, 548 of power series, 607 inflection point, 453 of Riemann, 510 injective function, A-7 partition, 488 integers, 3, A-2 properties of, 496, 523 integrable Riemann integrable, 510 by Cauchy’s second method, 504 Riemann’s criteria, 514 Riemann, 510 integral equation, 869, 870, 877, 925 integral integral form of the remainder, 470 absolutely convergent, 505, 508, 529 integration and uniform convergence, 569 by parts, 502 Cauchy’s first method, 489 by substitution, 502 Cauchy’s second method, 504 interchange of limit operations, 543 convergent, 504, 529 interior, 220 definite, 485 of a set, 229, 817 definition of for continuous functions, 489 point, 220, 817 divergent, 504 intermediate value property, 305 elliptic, 708 of continuous functions, 328 fundamental theorem of calculus, 499 of derivative, 444 generalized Riemann, 532 intersection of sets , A-3 improper Riemann, 528 inverse indefinite, 485 derivative of inverse function, 445 integrability of step functions, 520 function theorem, 790 integrable by Cauchy’s second method, 504 of a function, 416, A-8 integration by parts, 502 irrational number, 27 integration by substitution, 502 isolated point, 221 Lebesgue’s criteria, 518 in a metric space, 817 method of exhaustion, 486 isometry, 843 not the limit of the integral, 542 iteration, 34

Thomson*Bruckner*Bruckner Elementary Real Analysis, 2nd Edition (2008) A-38 ClassicalRealAnalysis.com Subject Index

IVP, 305, see intermediate value property of sequence, 87 lim sup Jacobian, 760 of function, 302 Jensen’s inequality, 454 of sequence, 87 jump discontinuity, 276, 295, 331 limit inferior, 87 Kasahara, 938 infinite, 277 kernel interchange of limit operations, 543 Dirichlet, 637 of a composition, 292 Fejér, 634 of a function, 264, 269, 272 Kuratowski, K., 940 of a function in Rn, 668 Kurzweil, J., 532 of a sequence, 42 L’Hôpital’srule, 434, 457, 460 right-hand, 274 Lagrange’s form of the remainder, 619 superior, 87 Lagrange, J., 467 uniqueness of, 44 Laplace’s equation, 693, 744 uniqueness of function limit, 279 limit point Laplacian, 745 n law of the mean, 430 in R , 652 least upper bound, 13 in a metric space, 817 Lebesgue Lindelöf’s theorem, 891 criteria for integrability, 518 Lindelöff’s covering theorem, 254 integral, 512 linear number, 891 equations, 868 Lebesgue, H., 517 function, 663 left continuous, 319 Liouville number, 926 left-hand derivative, 399 Liouville, J., 926 Leibniz rule, 703 Lipschitz Leibniz, G., 18 condition, 431, 879 lim inf constant, 879 of function, 302 function, 316, 431

Thomson*Bruckner*Bruckner Elementary Real Analysis, 2nd Edition (2008) Subject Index ClassicalRealAnalysis.com A-39

Lipschitz constant, 568 usual metric on R, 798 Lipschitz function, 568 metric space Lipschitz, R., 316 ε-net, 885 list, see set Bolzano-Weierstrass property, 883 local extrema and derivatives, 424 boundary point of a set, 817 locally bounded function, 240 bounded sequence, 822 lower bound, 11 bounded set, 817 lower sums, 490 Cantor intersection property, 854 Cantor space, 841 M-test, 553 Cauchy sequence, 850 magnification property of derivative, 405 closed ball, 816 mapping, 664, 765, A-10 closed set, 817 maximum, 11 closure, 817 Mazur, S., 356 compact, 883 Mazurkiewicz, S., 921 complete, 851 mean value theorem, 430 completeness proofs, 851 of Cauchy, 434 completion of, 854 second-order, 433 congruence, 843 measure zero set, 386 connected, 841 method continuous function, 828 of exhaustion, 486 contraction map, 859 of successive approximations, 863 convergence, 811 metric coordinate-wise convergence, 811 definition of, 798 definition of, 800 discrete metric, 799 dense set, 818 2 euclidean metric on R , 799 diameter, 817 n euclidean metric on R , 800 discrete metric, 799 Hausdorff metric, 810 distance between a point and a set, 818 on product space, 802 embedding, 844 property, 843 equicontinuity, 896 sup metric, 807

Thomson*Bruckner*Bruckner Elementary Real Analysis, 2nd Edition (2008) A-40 ClassicalRealAnalysis.com Subject Index

first category, 912 space [a, b] of continuous functions on [a, b], fixed point, 859 C806 function, 826 space ℓ of all bounded sequences, 805 ∞ function space, 806 space ℓ1 of all absolutely convergent series, 805 Hausdorff metric, 810 space M[a, b] of bounded functions on [a, b], 807 Hilbert cube, 813 subspace, 807 Hilbert space, 803 topological property, 836 homeomorphism, 836 topologically complete, 927 interior, 817 topologically equivalent, 836 interior point, 817 totally bounded, 892 isolated point, 817 transformation, 827 isometry, 843 uniform continuity, 887 limit point, 817 metric structure of reals, 22 neighborhood, 817 minimum, 11 nowhere dense, 818, 825, 906 mixed partials, 697 open ball, 816 modulus of continuity, 900 open cover, 889 monotone convergence theorem, 67 open set, 816 monotonic operator, 827 function, 333, 435 perfect set, 825 function, discontinuities of, 334 point of accumulation, 817 sequence, 66, 67 product, 802 subsequence, 79 remetrized, 927 Morgenstern, D., 617 residual, 912 second category, 912 natural numbers, 2, A-2 separable, 846 neighborhood, 219 Sierpinski’s space, 857 ε-neighborhood in a metric space, 817 space of nonempty closed subsets of [0, 1], deleted, 220 n K810 in R , 652 space c of all convergent sequences, 824 in a metric space, 817 relative, 313

Thomson*Bruckner*Bruckner Elementary Real Analysis, 2nd Edition (2008) Subject Index ClassicalRealAnalysis.com A-41 nested interval property, 71 set in a metric space, 816 Newton’s method, 29 operator, 664, 827 Newton, I., 29 order properties of sequence limits, 60 nonabsolute convergence, 136 order-preserving mapping, 366 nondecreasing ordered function, 333, 435 field, 10 sequence, 66 pairs, A-4 nonincreasing sum, 120 function, 333, 334, 435 orthogonal, 646 sequence, 66 orthogonality relations, 633 norm, 803 oscillation euclidean, 647 at a point, 379 triangle inequality for, 649 of a function, 378 nowhere dense, 354 on an interval, 378 in a metric space, 818, 906 nowhere differentiable function, 921 p-adic representation, 132 nowhere monotonic function, 920 parallelogram law, 651 partial derivative, 691 odd function, 615 cross, 697 one-to-one function, A-7 cross partial, 692 onto function, A-7 mixed, 697 open mixed partial, 692 ball, 816 partial order, 796 ball in Rn, 652 partial sums, 35 cover, 248, 889 partially ordered set, 795 cover in Rn, 680 partition, 246, 488, 511 interval, A-2 associated points, 489 map, 836 upper and lower sums, 490 relative to a set, 258 Peano space filling curve, 833 set, 229 Peano’s theorem, 901 set in Rn, 652 Peano, G., 931

Thomson*Bruckner*Bruckner Elementary Real Analysis, 2nd Edition (2008) A-42 ClassicalRealAnalysis.com Subject Index

perfect set, 363, 825 quotient of, 625 period of a sequence, 101 radius of convergence, 598 periodic sequence, 101 representation of a function, 605 perpetuity, 129 Taylor series, 615 Pfeffer, W. F., 532 uniform convergence of, 602 Picard’s theorem, 878 uniqueness of, 610 Poincaré,H., 583 preimage, A-11 point principle of induction, 19 boundary, 223 product interior, 220 dot product, 646 isolated, 221 finite, 111 of accumulation, 222, 817 infinite product, 201 pointwise metric, 802 bounded, 361 of absolutely convergent series, 184 convergence, 540 of derivatives not a derivative, 447 polynomial of metric spaces, 802 Chebychev polynomials, 639 of nonabsolutely convergent series, 186 Taylor polynomial, 618 of power series, 624 Pompeiu’s function, 585 of series, 183 Pompeiu, D., 584 proof positive integers, see natural numbers by contradiction, A-15 power rule for derivatives, 418 contraposition, A-17 power series, 171, 593, 594 converse, A-18 absolute convergence of, 598 counterexample, A-18, A-19 coefficients of, 594, 610 how to read, A-14 composition of, 628 how to write, A-15 continuity of, 606 indirect, A-15 derivative of, 609 induction, A-19 integral of, 607 reductio ad absurdum, A-15 products of, 624 what is, A-14

Thomson*Bruckner*Bruckner Elementary Real Analysis, 2nd Edition (2008) Subject Index ClassicalRealAnalysis.com A-43

why?, A-13 Riemann criteria for integrability, 514 quantiﬁer integrable, 510 , A-23 sums, 489 ∃, A-23 ∀ right continuous, 319 quotient right-hand derivative, 399 of power series, 625 right-hand limit, 274 rigid motion, 843 radius of convergence, 171, 598 Rolle’s theorem, 427 range of a function, A-7 Russell’s paradox, A-26 rational numbers, 3, A-2 Russell, B., A-26 are dense, 20 real line, 5 Saks, S., 473 real numbers, 4, A-2 second category set, 359, 912 as inﬁnite decimals, 5 second derivative, 401 rearrangement of a series, 173 second partial derivative, 691 rectangular sums, 198 separable space, 846 recursion formula, 33 separate a set, 684 reductio ad absurdum, A-15 separately continuous, 832 regular summability method, 191 separation of compact sets, 259 relation, A-5 sequence equivalence relation, A-12 absolute values, 64 remainder algebraic properties of limits, 53 integral form of, 470 bounded, 49 Lagrange form of, 468 bounded variation, 138 Lagrange’s form of, 619 Cauchy criterion, 84 Taylor, 618 Cauchy sequence, 85 remetrized, 927 Cauchy sequence in a metric space, 850 removable discontinuity, 331 cluster point, 83 representation of a function by a power series, 605 contractive, 97 residual set, 359, 912 convergent subsequence, 80

Thomson*Bruckner*Bruckner Elementary Real Analysis, 2nd Edition (2008) A-44 ClassicalRealAnalysis.com Subject Index

converges, 41 tail, 46 converging in a metric space, 811 terms of, 32 decreasing, 66 uniqueness of limits, 44 definition of, 32 series, 120 divergent to infinity, 47 p-harmonic series, 213 diverges, 42 absolute convergence, 136 Fibonacci sequence, 37, 97 alternating harmonic series, 125 in Rn, 656 binomial series, 161 increasing, 66 boundedness criterion, 132 lim inf, 87 Cauchy criterion, 133 lim sup, 87 Cesàromethod, 190 limit inferior, 87 comparison with series, 177 limit of, 42 comparison with sums, 177 limit superior, 87 convergent, 121 maxima and minima, 64 divergent, 121 monotone convergence theorem, 67 formula for sum of geometric series, 124 monotonic, 66 Fourier series, 629 monotonic subsequence, 79 geometric, 124 nondecreasing, 66 harmonic series, 125 nonincreasing, 66 nonabsolute convergence, 136 of functions uniformly bounded, 562, 574 of functions, 538 of functions uniformly Cauchy, 551 power series, 171, 594 of functions uniformly convergent, 549 product of, 183 of partial sums, 35 product of absolutely convergent series, 184 of real numbers, 32 product of nonabsolutely convergent series, 186 order properties of limits, 60 rearrangement, 173 periodic, 101 summability methods, 189 recursion formula for, 33 tail, 123 squeeze property of limits, 62 Taylor series, 615 subsequence, 78 telescoping, 123

Thomson*Bruckner*Bruckner Elementary Real Analysis, 2nd Edition (2008) Subject Index ClassicalRealAnalysis.com A-45

trigonometric series, 209, 629 dense in a metric space, 818 unconditionally convergent, 174, 176 dense in itself in a metric space, 825 uniqueness of sum, 122 dense set, 20, 352 set derived, 226 set, 375 diameter of in a metric space, 817 F σ δ set, 373 difference of sets, A-3 Gas a list, A-3 disconnected, 260 Borel, 372 distance from set to a point, 818 boundary point of in a metric space, 817 empty set , A-2 bounded, 11 finite intersection∅ property, 259 bounded closed set in a metric space, 817 first category, 359, 912 bounded set in Rn, 653 homeomorphic sets, 366 Cantor set, 365, 366, 370, 921 inductive set, 26 Cantor ternary set, 239, 363, 365, 368 interior, 817 Cartesian product, A-5 interior of, 229 closed, 227 interior point, 817 closed interval, A-2 intersection of sets, A-3 closed relative to a set, 258 measure zero, 386 closed set in Rn, 652 member of, A-2 closed set in a metric space, 817 not a member of, A-2 closure of a set, 228 nowhere dense, 354 closure of in a metric space, 817 nowhere dense in a metric space, 818 co-countable, 257 nowhere dense set in a metric space, 825 compact, 252, 883 of integers, A-2 compact in Rn, 680 of natural numbers, A-2 components of open set, 230 of ordered pairs, A-4 connected, 260 of points of continuity of a function, 382 countable, 37, 39, 256 of rational numbers, A-2 De Morgan’s laws, A-4 of real numbers, A-2 definition of, A-1 of uniqueness, 217

Thomson*Bruckner*Bruckner Elementary Real Analysis, 2nd Edition (2008) A-46 ClassicalRealAnalysis.com Subject Index

of zero content, 392 subspace, 807 open, 229 vector space, 645 open interval, A-2 square sums, 198 open relative to a set, 258 squeeze property open set in Rn, 652 for function limits, 288 open set in a metric space, 816 of sequence limits, 62 perfect, 363 step function, 298 perfect set in a metric space, 825 integrability of, 520 properties of closed sets, 237 straddled derivative, 471 properties of open sets, 236 strategy in the Banach-Mazur game, 358 quantifier, A-23 strictly monotonic quantifier , A-23 function, 333 quantifier ∃, A-23 sequence, 67 relation, A-5∀ subfield, 8 residual, 359, 912 subgroup of real numbers, 27 second category, 359, 912 subsequence, 78 set-builder notation, A-3 monotonic, 79 subset, A-3 subset, A-3 uncountable, 256 subspace, 807 union of sets, A-3 of a complete metric space, 853 set-builder notation, A-3 successive approximations, 863 Sierpinski’s space, 857 sum smooth function, 433 Cauchy criterion for unordered sum, 115 space convergent unordered sum, 113 function space, 806 in closed form, 106 Hilbert space, 686, 803 telescoping, 107 metric space, 800 unordered, 112 space ℓ of all bounded sequences, 805 summability method, 189 ∞ space ℓ1 of all absolutely convergent series, 805 Abel, 193 space c of all convergent sequences, 824 Cesàro, 190

Thomson*Bruckner*Bruckner Elementary Real Analysis, 2nd Edition (2008) Subject Index ClassicalRealAnalysis.com A-47

regular, 191 tests summation by parts, 108 Abel’s test, 165 sums alternating series test, 162 circular sums, 198 condensation test, 150 finite sums, 105 direct comparison test, 140 rectangular sums, 198 Dirichlet’s test, 164 square sums, 198 for convergence of series, 140 sup, see supremum Gauss’s test, 158 sup metric, 807 integral test, 153 supremum, 13 Kummer’s test, 154 surjective function, A-7 limit comparison test, 143 symmetric derivative, 402 Raabe’s test, 157 system of linear equations, 868 ratio comparison test, 145 ratio test, 146 tail root test, 149 of a sequence, 46 trivial test, 140 of a series, 123 theorem Tauberian theorem, 197 Baire category theorem, 359, 911 Taylor Banach fixed point theorem, 861 integral form of the remainder for Taylor series, Bernstein, 623 470 Bolzano-Weierstrass, 80, 242 Lagrange form of the remainder for Taylor in Rn, 659 series, 468 Cantor intersection theorem, 244 polynomial, 467, 468, 618 collage theorem, 873 remainder, 618 Denjoy-Young-Saks, 473 series, 615 fundamental theorem of calculus, 499, 525 telescoping series, 123 Heine-Borel, 248 telescoping sum, 107 inverse function theorem, 790 ternary L’Hôpital’srule, 457, 460 expansion, 261 Lagrange, 467 representation of Cantor set, 367

Thomson*Bruckner*Bruckner Elementary Real Analysis, 2nd Edition (2008) A-48 ClassicalRealAnalysis.com Subject Index

Lindelöff, 254 type δ, 373 mean value theorem, 430 G monotone convergence theorem, 67 unbounded above, 11 of Abel, 187, 193 unconditionally convergent, 174, 176 of Arzelàand Ascoli, 897 uncountable set, 256 of Cantor, 38 uniform convergence, 549 of Dini, 566 Abel’s test, 555 of Dirichlet, 174 and continuity, 564 of Fejér, 634 and derivatives, 578 of Lindelöf, 891 and the integral, 569 of Peano, 901 as convergence in the space M[a, b], 811 of Picard, 878 Cauchy criterion, 551 of Riemann, 176 Fourier series, 630 of Rolle, 427 of a power series, 602 squeeze theorem, 62 Weierstrass M-test, 553 squeeze theorem for function limits, 288 uniformly bounded family of functions, 361, 562, Urysohn’s lemma, 927 574, 896 Weierstrass approximation theorem, 638 uniformly Cauchy, 551 topological property, 836 uniformly continuous, 322 topologically complete metric space, 927 in a metric space, 887 n topologically equivalent, see homeomorphic on R , 681 topology, 236 union of sets , A-3 in a metric space, 836 uniqueness totally bounded, 892, 894 of function limit, 279 transcendental number, 41 of power series, 610 transformation, 664, 765, 827 of sequence limits, 44 trapping principle for derivatives, 400 of sum of series, 122 triangle inequality, 24, 649 unstraddled derivative, 471 trigonometric series, 209, 629 upper bound, 11 upper sums, 490 type σ, 375 F Urysohn’s lemma, 927

Thomson*Bruckner*Bruckner Elementary Real Analysis, 2nd Edition (2008) Subject Index ClassicalRealAnalysis.com A-49

Urysohn, P., 927 vector space, 645 product, 684 vector-valued function, 664 Volterra, V., 584 weak contraction, 888 Weierstrass M-test, 553 approximation theorem, 638 Weierstrass, K.T.W, 80 well ordering of IN, 19 Wronski, J. de, 483 Wronskian, 472

Young, G., 473 zero content, 392 zero measure, 386

Thomson*Bruckner*Bruckner Elementary Real Analysis, 2nd Edition (2008)