lemma A.488 Small chords cut off small arcs

open in the book · appendices/A-long-proofs.tex:23924 · p. 3032

Rests on

depends_on theorem 7.25 Heine–Cantor: uniform continuity ¶
- depends_on proposition 7.22 Sequential characterization ¶
  - depends_on definition 7.20 Continuity at a point ¶
    - depends_on definition 7.16 Limit ¶
      - depends_on definition 7.2 Absolute value ¶
      - depends_on definition 7.9 Real function ¶
    - depends_on equation 7.7 eq:ana-limit-left ¶
    - depends_on equation 7.5 eq:ana-limit-right ¶
  - depends_on definition 7.4 Convergence ¶
    - depends_on definition 7.2 Absolute value ¶ ↺
  - proves proof ch:05-real-analysis@proof-7 ¶
- depends_on theorem 7.7 Bolzano–Weierstrass ¶
  - depends_on corollary A.45 Archimedean property and density of $\Q$ ¶
    - depends_on definition A.36 Cut ¶
    - 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-32 ¶
  - depends_on corollary A.46 Monotone convergence ¶
    - depends_on theorem A.40 Least-upper-bound property ¶ ↺
    - proves proof app:A-long-proofs@proof-33 ¶
  - proves proof ch:05-real-analysis@proof-4 ¶
- proves proof ch:05-real-analysis@proof-10 ¶
depends_on theorem 6.31 Compactness and sequential compactness ¶
- depends_on definition 6.9 Compact set ¶
  - depends_on definition 6.5 Open cover ¶
    - depends_on definition 6.2 Open set ¶
      - depends_on definition 6.1 Topological space ¶
    - depends_on equation 3.51 eq:set-indexed ¶
- depends_on definition 6.29 Sequential compactness ¶
  - depends_on definition 6.24 Metric ¶
    - depends_on definition 3.43 Map ¶
      - depends_on definition 3.24 Quantifiers ¶
      - depends_on definition 3.28 Set ¶
    - depends_on definition 3.38 Ordered pair and Cartesian product ¶
  - depends_on definition 6.27 Convergence; Cauchy sequence; completeness ¶
    - depends_on definition 6.24 Metric ¶ ↺
- depends_on lemma 6.30 Lebesgue number ¶
  - depends_on definition 6.26 Open ball; metric topology ¶
    - depends_on definition 6.24 Metric ¶ ↺
    - depends_on definition 6.2 Open set ¶ ↺
    - depends_on definition 6.1 Topological space ¶ ↺
  - depends_on definition 6.5 Open cover ¶ ↺
  - depends_on definition 6.29 Sequential compactness ¶ ↺
  - proves proof ch:04-topology@proof-10 ¶
- proves proof ch:04-topology@proof-11 ¶
proves proof app:A-long-proofs@proof-289 ¶

Supports

depends_on theorem A.490 Equicontinuity of the normalised class ¶

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`	→	Heine–Cantor: uniform continuity	declared	appendices/A-long-proofs.tex:23931
`depends_on`	→	Compactness and sequential compactness	declared	appendices/A-long-proofs.tex:23931
`depends_on`	←	Equicontinuity of the normalised class	declared	appendices/A-long-proofs.tex:24032
`proves`	←	app:A-long-proofs@proof-289	declared	appendices/A-long-proofs.tex:23934