Math 2.0

Sequences, series, summability Proof

Proof that the Cauchy product of absolutely convergent series converges absolutely to the product

Builds on Absolute convergenceCauchy productPartial sumA series of nonnegative terms converges iff its partial sums are boundedLimit of a product of sequencesLimit of a sum of sequencesSqueeze theoremEpsilon-N definition of the limit of a sequenceTriangle inequality

Proof

We compare the partial sums of the Cauchy product with products of partial sums.

  1. Let ∑an\sum a_n and ∑bn\sum b_n converge absolutely, with sums AA and BB. Let cn=∑k=0nakbn−kc_n = \sum_{k=0}^{n} a_k b_{n-k} define the Cauchy product.
  2. Let ANA_N, BNB_N, CNC_N be the partial sums of ∑an\sum a_n, ∑bn\sum b_n, ∑cn\sum c_n. Let PN=∑n≤N∣an∣P_N = \sum_{n \le N} \abs{a_n} and QN=∑n≤N∣bn∣Q_N = \sum_{n \le N} \abs{b_n}.
  3. Let P=∑∣an∣P = \sum \abs{a_n} and Q=∑∣bn∣Q = \sum \abs{b_n}. The terms are nonnegative, so PN≤PP_N \le P and QN≤QQ_N \le Q by the criterion for nonnegative series.
  4. The term cnc_n is the sum of aibja_i b_j over the pairs (i,j)(i, j) of natural numbers with i+j=ni + j = n. So CNC_N is the sum of aibja_i b_j over the set TNT_N of pairs with i+j≤Ni + j \le N.
  5. Each pair in TNT_N has i≤Ni \le N and j≤Nj \le N. By the triangle inequality, ∑n=0N∣cn∣≤∑(i,j)∈TN∣ai∣∣bj∣≤∑i=0N∑j=0N∣ai∣∣bj∣=PNQN≤PQ.\sum_{n=0}^{N} \abs{c_n} \le \sum_{(i,j) \in T_N} \abs{a_i} \abs{b_j} \le \sum_{i=0}^{N} \sum_{j=0}^{N} \abs{a_i} \abs{b_j} = P_N Q_N \le P Q.
  6. So the partial sums of ∑∣cn∣\sum \abs{c_n} are bounded above by PQPQ. By the criterion for nonnegative series, ∑∣cn∣\sum \abs{c_n} converges. Hence ∑cn\sum c_n converges absolutely.
  7. Let UNU_N be the set of pairs (i,j)(i, j) with i≤Ni \le N, j≤Nj \le N and i+j>Ni + j > N. Then ANBN=∑i,j≤Naibj=CN+∑(i,j)∈UNaibjA_N B_N = \sum_{i, j \le N} a_i b_j = C_N + \sum_{(i,j) \in U_N} a_i b_j.
  8. Let EN=∑(i,j)∈UN∣ai∣∣bj∣E_N = \sum_{(i,j) \in U_N} \abs{a_i} \abs{b_j}. By step 7 and the triangle inequality, ∣ANBN−CN∣≤EN\abs{A_N B_N - C_N} \le E_N.
  9. In the same way as step 7, PNQN=∑(i,j)∈TN∣ai∣∣bj∣+ENP_N Q_N = \sum_{(i,j) \in T_N} \abs{a_i} \abs{b_j} + E_N.
  10. Let h=⌊N/2⌋h = \floor{N/2}. If i≤hi \le h and j≤hj \le h, then i+j≤Ni + j \le N. So ∑(i,j)∈TN∣ai∣∣bj∣≥PhQh\sum_{(i,j) \in T_N} \abs{a_i} \abs{b_j} \ge P_h Q_h.
  11. By steps 9 and 10, 0≤EN≤PNQN−P⌊N/2⌋Q⌊N/2⌋0 \le E_N \le P_N Q_N - P_{\floor{N/2}} Q_{\floor{N/2}}.
  12. By the product rule for limits, PNQN→PQP_N Q_N \to PQ. Let ε>0\varepsilon > 0, and take KK with ∣PkQk−PQ∣<ε\abs{P_k Q_k - PQ} < \varepsilon for k≥Kk \ge K, by the epsilon-N form.
  13. For N≥2KN \ge 2K we have ⌊N/2⌋≥K\floor{N/2} \ge K. So P⌊N/2⌋Q⌊N/2⌋→PQP_{\floor{N/2}} Q_{\floor{N/2}} \to PQ as N→∞N \to \infty.
  14. By the sum rule for limits, the right side of step 11 tends to 00. By the squeeze theorem, EN→0E_N \to 0.
  15. By step 8, −EN≤ANBN−CN≤EN-E_N \le A_N B_N - C_N \le E_N. The squeeze theorem gives ANBN−CN→0A_N B_N - C_N \to 0.
  16. By the product rule, ANBN→ABA_N B_N \to AB. So CN=ANBN−(ANBN−CN)→AB−0=ABC_N = A_N B_N - (A_N B_N - C_N) \to AB - 0 = AB by the sum rule.
  17. Hence ∑cn\sum c_n converges absolutely, and its sum is ABAB.

Idea

The product ANBNA_N B_N sums aibja_i b_j over a square of indices, while CNC_N sums over the triangle i+j≤Ni + j \le N inside it. The missing corner is controlled by the same corner for the absolute values. That corner lies between the squares of sides N/2N/2 and NN, so it tends to 00.

Reading path

To understand this page from the ground up, read these 66 pages in order. The foundations come first. The path takes the proofs Proof of the limit of a sum of sequences, Proof of the approximation property of the supremum, Proof that the reals form an ordered field, with the Cauchy sequence construction, Proof of the limit of a product of sequences, Proof that the absolute value is multiplicative, Proof of the squeeze theorem for sequences, Proof of the triangle inequality for real numbers, Proof of the epsilon-N definition of the limit of a sequence, 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. Element definition
  3. Integers object
  4. Mathematical induction concept
  5. Ordered field definition
  6. Proposition concept
  7. Rational numbers object
  8. Real numbers object
  9. Sequence definition
  10. Set bounded above definition
  11. Set definition
  12. Ring of integers statement
  13. Implication definition
  14. Predicate definition
  15. The reals form a field statement
  16. Real sequence definition
  17. Supremum definition
  18. Ordered pair definition
  19. Universal quantifier definition
  20. The rationals form a field statement
  21. Bounded sequence definition
  22. Limit of a sequence definition
  23. Monotone sequence definition
  24. Series definition
  25. Completeness axiom statement
  26. Cartesian product definition
  27. Subset definition
  28. The rationals form an ordered field statement
  29. Convergent sequence definition
  30. Cauchy product definition
  31. Partial sum definition
  32. Binary relation definition
  33. Order on the reals definition
  34. Convergent series definition
  35. Antisymmetric relation definition
  36. Reflexive relation definition
  37. Transitive relation definition
  38. Sum of a series definition
  39. Absolute convergence definition
  40. Preorder definition
  41. Partial order definition
  42. Total order definition
  43. Proof that the reals form an ordered field, with the Cauchy sequence construction proof
  44. The reals form an ordered field statement
  45. Absolute value definition
  46. Proof of the approximation property of the supremum proof
  47. Proof of the triangle inequality for real numbers proof
  48. Proof that the absolute value is multiplicative proof
  49. Proof of the epsilon-N definition of the limit of a sequence proof
  50. Approximation property of the supremum statement
  51. Triangle inequality statement
  52. Absolute value is multiplicative statement
  53. Epsilon-N definition of the limit of a sequence statement
  54. Proof of the limit of a sum of sequences proof
  55. Proof of the squeeze theorem for sequences proof
  56. Proof that convergent sequences are bounded proof
  57. Proof of the monotone convergence theorem for sequences proof
  58. Limit of a sum of sequences statement
  59. Convergent sequences are bounded statement
  60. Squeeze theorem statement
  61. Monotone convergence theorem for sequences statement
  62. Proof of the limit of a product of sequences proof
  63. Proof that a series of nonnegative terms converges iff its partial sums are bounded proof
  64. Limit of a product of sequences statement
  65. A series of nonnegative terms converges iff its partial sums are bounded statement
  66. Proof that the Cauchy product of absolutely convergent series converges absolutely to the product proof

Where this leads

425 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Cauchy_product, Convergence and Mertens' theorem
About this page
TagYP5Q4Z9ZTZ
URIm2://YP5Q4Z9ZTZ
Kindproof
WritingW0 stub
MSC40A05
ProvesCauchy product of absolutely convergent series
UsesAbsolute convergence, Cauchy product, Partial sum, A series of nonnegative terms converges iff its partial sums are bounded, Limit of a product of sequences, Limit of a sum of sequences, Squeeze theorem, Epsilon-N definition of the limit of a sequence, Triangle inequality
Changed4 November 2026

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