lemma A.290 $h$ is Lipschitz

open in the book · appendices/A-long-proofs.tex:14543 · p. 2935

Rests on

depends_on lemma A.288 The iteration converges ¶
- depends_on lemma A.287 The Newton map contracts ¶
  - depends_on definition 7.98 Functions of class $C^{1}$ ¶
    - depends_on definition 7.20 Continuity at a point ¶
      - depends_on definition 7.16 Limit ¶
      - depends_on equation 7.7 eq:ana-limit-left ¶
      - depends_on equation 7.5 eq:ana-limit-right ¶
    - depends_on definition 7.97 Partial derivative; gradient ¶
      - depends_on definition 7.26 Derivative of a function at a point ¶
      - depends_on definition 5.15 Basis ¶
  - depends_on equation A.523 eq:app-implicit-hypothesis ¶
  - depends_on theorem 7.35 Mean value theorem ¶
    - depends_on proposition 7.29 Linearity ¶
      - depends_on definition 7.11 Sum of functions ¶
      - depends_on equation 7.14 eq:ana-derivh ¶
      - depends_on proposition 7.6 Algebra of limits ¶
      - proves proof ch:05-real-analysis@proof-13 ¶
    - depends_on theorem 7.34 Rolle ¶
      - depends_on lemma 7.33 Fermat: interior extremum ¶
      - depends_on theorem 7.24 Extreme value theorem ¶
      - proves proof ch:05-real-analysis@proof-18 ¶
    - proves proof ch:05-real-analysis@proof-19 ¶
  - proves proof app:A-long-proofs@proof-180 ¶
- depends_on proposition 7.46 Geometric series ¶
  - 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 $\R$ ¶
      - 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 ¶ ↺
  - proves proof ch:05-real-analysis@proof-27 ¶
- depends_on theorem 7.8 Cauchy criterion ¶
  - depends_on corollary A.47 Cauchy completeness ¶
    - depends_on corollary A.46 Monotone convergence ¶ ↺
    - depends_on theorem A.40 Least-upper-bound property ¶ ↺
    - proves proof app:A-long-proofs@proof-34 ¶
  - depends_on definition 7.4 Convergence ¶ ↺
  - proves proof ch:05-real-analysis@proof-5 ¶
- proves proof app:A-long-proofs@proof-181 ¶
depends_on theorem 7.35 Mean value theorem ¶ ↺
proves proof app:A-long-proofs@proof-183 ¶

Supports

depends_on lemma A.291 $h$ is differentiable, with the stated derivative ¶

Neighborhood

Every logical edge within two steps of this node.

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partly declared
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Edges

type	direction	node	provenance	where
`depends_on`	→	The iteration converges	declared	appendices/A-long-proofs.tex:14549
`depends_on`	→	Mean value theorem	declared	appendices/A-long-proofs.tex:14549
`depends_on`	←	$h$ is differentiable, with the stated derivative	declared	appendices/A-long-proofs.tex:14587
`proves`	←	app:A-long-proofs@proof-183	declared	appendices/A-long-proofs.tex:14552