Math 2.0

Sequences, series, summability Proof

Proof that absolutely convergent series converge

Builds on Absolute convergenceCauchy criterion for seriesTriangle inequalityA series of nonnegative terms converges iff its partial sums are boundedLimits preserve non-strict inequalities

Proof

We use the Cauchy criterion for series.

  1. By the definition of absolute convergence, the series ∑∣an∣\sum \abs{a_n} converges. Let TT be its sum.
  2. Let ε>0\varepsilon > 0. By the Cauchy criterion for series applied to ∑∣an∣\sum \abs{a_n}, there is NN with ∑k=m+1n∣ak∣<ε\sum_{k=m+1}^{n} \abs{a_k} < \varepsilon for all n>m≥Nn > m \ge N.
  3. For such mm and nn, the triangle inequality, applied repeatedly, gives ∣∑k=m+1nak∣≤∑k=m+1n∣ak∣<ε\abs{\sum_{k=m+1}^{n} a_k} \le \sum_{k=m+1}^{n} \abs{a_k} < \varepsilon.
  4. By the Cauchy criterion for series again, ∑an\sum a_n converges. Let SS be its sum and sN=∑n=0Nans_N = \sum_{n=0}^{N} a_n.
  5. Let tN=∑n=0N∣an∣t_N = \sum_{n=0}^{N} \abs{a_n}. By the criterion for nonnegative series, T=sup⁡NtNT = \sup_N t_N, so tN≤Tt_N \le T for all NN.
  6. By the triangle inequality, ∣sN∣≤tN≤T\abs{s_N} \le t_N \le T. So −T≤sN≤T-T \le s_N \le T for every NN.
  7. Since sN→Ss_N \to S and limits preserve non-strict inequalities, −T≤S≤T-T \le S \le T.
  8. So ∣∑n=0∞an∣=∣S∣≤T=∑n=0∞∣an∣\abs{\sum_{n=0}^{\infty} a_n} = \abs{S} \le T = \sum_{n=0}^{\infty} \abs{a_n}.

Idea

A block of terms am+1+⋯+ana_{m+1} + \cdots + a_n is never larger in absolute value than the same block of absolute values. If the blocks of absolute values become small, so do the original blocks, and the Cauchy criterion gives convergence.

Reading path

To understand this page from the ground up, read these 103 pages in order. The foundations come first. The path takes the proofs Proof of the approximation property of the supremum, Proof that every sequence has a monotone subsequence, by peaks, Proof of the commutativity of addition by induction, Proof that convergent sequences are Cauchy, Proof of the Cauchy criterion for series, Proof of the Cauchy criterion for sequences by the Bolzano–Weierstrass theorem, Proof that the reals form an ordered field, with the Cauchy sequence construction, Proof of the double negation law by a truth table, Proof of strong induction from ordinary induction, Proof that Cauchy sequences are bounded, Proof of the triangle inequality for real numbers, Proof of the epsilon-N definition of the limit of a sequence, Proof that limits preserve non-strict inequalities, Proof of the well-ordering principle from strong induction, Proof of the Bolzano–Weierstrass theorem by a monotone subsequence, Proof that a decreasing sequence bounded below converges, Proof that convergent sequences are bounded, Proof of the monotone convergence theorem for sequences, Proof that a series of nonnegative terms converges iff its partial sums are bounded.

  1. Construction of the reals from Cauchy sequences concept
  2. Definition by recursion concept
  3. Element definition
  4. Integers object
  5. Mathematical induction concept
  6. Natural numbers object
  7. Ordered field definition
  8. Peano axioms concept
  9. Proposition concept
  10. Rational numbers object
  11. Real numbers object
  12. Sequence definition
  13. Set bounded above definition
  14. Set definition
  15. Ring of integers statement
  16. Successor function definition
  17. Conjunction definition
  18. Disjunction definition
  19. Implication definition
  20. Negation definition
  21. Predicate definition
  22. The reals form a field statement
  23. Real sequence definition
  24. Subsequence definition
  25. Infimum definition
  26. Supremum definition
  27. Ordered pair definition
  28. Addition of natural numbers definition
  29. Biconditional definition
  30. The successor function is injective statement
  31. Truth table concept
  32. Zero is not a successor statement
  33. Modus ponens concept
  34. The rationals form a field statement
  35. Universal quantifier definition
  36. Bounded sequence definition
  37. Cauchy sequence definition
  38. Limit of a sequence definition
  39. Monotone sequence definition
  40. Series definition
  41. Completeness axiom statement
  42. Cartesian product definition
  43. Proof of the commutativity of addition by induction proof
  44. Order on the natural numbers definition
  45. Subset definition
  46. Propositional logic concept
  47. Mathematical proof concept
  48. The rationals form an ordered field statement
  49. Convergent sequence definition
  50. Partial sum definition
  51. Binary relation definition
  52. Commutativity of addition statement
  53. Proof of strong induction from ordinary induction proof
  54. Order on the reals definition
  55. Tautology definition
  56. Convergent series definition
  57. Antisymmetric relation definition
  58. Reflexive relation definition
  59. Transitive relation definition
  60. Strong induction statement
  61. Logical equivalence definition
  62. Sum of a series definition
  63. Absolute convergence definition
  64. Preorder definition
  65. Proof of the well-ordering principle from strong induction proof
  66. Proof of the double negation law by a truth table proof
  67. Partial order definition
  68. Well-ordering principle statement
  69. Double negation law statement
  70. Total order definition
  71. Proof by contradiction concept
  72. Proof that every sequence has a monotone subsequence, by peaks proof
  73. Proof that the reals form an ordered field, with the Cauchy sequence construction proof
  74. Every sequence has a monotone subsequence statement
  75. The reals form an ordered field statement
  76. Absolute value definition
  77. Proof of the approximation property of the supremum proof
  78. Proof of the triangle inequality for real numbers proof
  79. Proof of the epsilon-N definition of the limit of a sequence proof
  80. Approximation property of the supremum statement
  81. Triangle inequality statement
  82. Epsilon-N definition of the limit of a sequence statement
  83. Proof that limits preserve non-strict inequalities proof
  84. Proof that convergent sequences are bounded proof
  85. Proof that convergent sequences are Cauchy proof
  86. Proof that Cauchy sequences are bounded proof
  87. Proof of the monotone convergence theorem for sequences proof
  88. Limits preserve non-strict inequalities statement
  89. Convergent sequences are bounded statement
  90. Convergent sequences are Cauchy statement
  91. Cauchy sequences are bounded statement
  92. Monotone convergence theorem for sequences statement
  93. Proof that a series of nonnegative terms converges iff its partial sums are bounded proof
  94. Proof that a decreasing sequence bounded below converges proof
  95. Decreasing sequence bounded below converges statement
  96. A series of nonnegative terms converges iff its partial sums are bounded statement
  97. Proof of the Bolzano–Weierstrass theorem by a monotone subsequence proof
  98. Bolzano–Weierstrass theorem statement
  99. Proof of the Cauchy criterion for sequences by the Bolzano–Weierstrass theorem proof
  100. Cauchy criterion for sequences statement
  101. Proof of the Cauchy criterion for series proof
  102. Cauchy criterion for series statement
  103. Proof that absolutely convergent series converge proof

Where this leads

498 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Absolute_convergence, Alternative proof using the Cauchy criterion and triangle inequality
About this page
Tag5VRNB5BEET
URIm2://5VRNB5BEET
Kindproof
WritingW0 stub
MSC40A05
ProvesAbsolutely convergent series converge
UsesAbsolute convergence, Cauchy criterion for series, Triangle inequality, A series of nonnegative terms converges iff its partial sums are bounded, Limits preserve non-strict inequalities
Changed4 November 2026

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