proposition 7.49 Cauchy product

open in the book · parts/02-mathematical-methods/05-real-analysis.tex:945 · p. 218

Rests on

depends_on corollary A.46 Monotone convergence ¶
- depends_on theorem A.40 Least-upper-bound property ¶
  - depends_on definition A.36 Cut ¶
  - depends_on definition A.38 Order on $\R$ ¶
    - depends_on definition A.36 Cut ¶ ↺
  - proves proof app:A-long-proofs@proof-29 ¶
- proves proof app:A-long-proofs@proof-33 ¶
depends_on definition 7.45 Series ¶
- depends_on definition 7.2 Absolute value ¶
- depends_on definition 7.4 Convergence ¶
  - depends_on definition 7.2 Absolute value ¶ ↺
depends_on proposition 7.6 Algebra of limits ¶
- depends_on definition 7.4 Convergence ¶ ↺
- depends_on proposition 7.3 Triangle inequality ¶
  - depends_on definition 7.2 Absolute value ¶ ↺
  - proves proof ch:05-real-analysis@proof-1 ¶
- proves proof ch:05-real-analysis@proof-3 ¶
proves proof ch:05-real-analysis@proof-30 ¶

Supports

depends_on lemma 7.60 Functional equation of the exponential ¶
- depends_on proposition 7.68 The natural base ¶
  - depends_on remark 7.70 What the condition says ¶
- depends_on proposition 7.61 The logarithm ¶
  - depends_on corollary 7.69 $\ee$ is the unique self-reproducing base ¶
    - depends_on remark 7.70 What the condition says ¶ ↺
  - depends_on definition 7.62 Real powers ¶
    - depends_on definition 7.90 The golden ratio ¶
      - depends_on proposition 7.91 The golden ratio ¶
    - depends_on proposition 7.63 Laws of real powers ¶
      - depends_on definition 7.90 The golden ratio ¶ ↺
      - depends_on lemma 7.67 The limit over real exponents ¶
      - depends_on proposition 7.68 The natural base ¶ ↺
      - depends_on proposition 7.91 The golden ratio ¶ ↺
      - depends_on proposition 7.89 Irrationality of square roots ¶
    - depends_on proposition 7.89 Irrationality of square roots ¶ ↺
  - depends_on lemma 7.67 The limit over real exponents ¶ ↺
  - depends_on proposition 7.68 The natural base ¶ ↺
  - depends_on proposition 7.63 Laws of real powers ¶ ↺
- depends_on proposition 7.63 Laws of real powers ¶ ↺
depends_on lemma 7.71 Derivatives; the Pythagorean identity ¶
- depends_on lemma 7.75 The first quadrant ¶
  - depends_on proposition 7.76 $\pi$ as the circle constant ¶
  - depends_on proposition 7.77 Special values, periodicity, and the kernel ¶
    - depends_on lemma 7.78 Chord, arc, tangent ¶
      - depends_on proposition 7.79 The polygon recursion ¶
    - depends_on proposition 7.85 Irrationality of $\pi$ ¶
      - depends_on remark 7.94 Three numbers, two kinds ¶
      - depends_on remark 7.87 Priority, and what irrationality does not give ¶
    - depends_on proposition 7.79 The polygon recursion ¶ ↺
    - depends_on proposition 7.83 Viète's product ¶
      - depends_on remark 7.84 One construction, two faces ¶
    - depends_on proposition 125.2 Crystallographic restriction ¶
      - depends_on remark 125.3 What the theorem does and does not forbid ¶
    - depends_on remark 7.88 $\pi$ elsewhere in this treatise ¶
  - depends_on proposition 125.2 Crystallographic restriction ¶ ↺
- depends_on lemma 7.78 Chord, arc, tangent ¶ ↺
- depends_on lemma 9.136 Wallis integrals ¶
  - depends_on proposition 9.139 Series for the complete integral of the first kind ¶
    - depends_on proposition 9.144 The pendulum equation at finite amplitude ¶
- depends_on proposition 7.76 $\pi$ as the circle constant ¶ ↺
- depends_on proposition 7.85 Irrationality of $\pi$ ¶ ↺
- depends_on proposition 125.2 Crystallographic restriction ¶ ↺
- depends_on proposition 9.141 First properties ¶
  - depends_on proposition 9.143 A cubic quadrature between two turning points ¶
  - depends_on proposition 9.142 The differential equation of the sine amplitude ¶
- depends_on remark 7.80 Why these are the polygon perimeters ¶
depends_on lemma 7.72 Addition theorems ¶
- depends_on proposition 7.79 The polygon recursion ¶ ↺
- depends_on proposition 7.77 Special values, periodicity, and the kernel ¶ ↺
- depends_on proposition 7.83 Viète's product ¶ ↺
- depends_on proposition 125.2 Crystallographic restriction ¶ ↺
- depends_on remark 7.88 $\pi$ elsewhere in this treatise ¶ ↺
- depends_on remark 7.80 Why these are the polygon perimeters ¶ ↺

Neighborhood

Every logical edge within two steps of this node.

declared and complete
partly declared
a check failed
not graded
declared in the source
inferred from structure

Edges

type	direction	node	provenance	where
`depends_on`	→	Monotone convergence	declared	parts/02-mathematical-methods/05-real-analysis.tex:953
`depends_on`	→	Series	declared	parts/02-mathematical-methods/05-real-analysis.tex:953
`depends_on`	→	Algebra of limits	declared	parts/02-mathematical-methods/05-real-analysis.tex:953
`depends_on`	←	Functional equation of the exponential	declared	parts/02-mathematical-methods/05-real-analysis.tex:1361
`depends_on`	←	Derivatives; the Pythagorean identity	declared	parts/02-mathematical-methods/05-real-analysis.tex:1765
`depends_on`	←	Addition theorems	declared	parts/02-mathematical-methods/05-real-analysis.tex:1834
`proves`	←	ch:05-real-analysis@proof-30	declared	parts/02-mathematical-methods/05-real-analysis.tex:956