lemma A.176 Helly–Bray

open in the book · appendices/A-long-proofs.tex:9130 · p. 2877

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 A.173 Helly's selection theorem ¶
- depends_on definition 11.21 Distribution function ¶
  - depends_on definition 11.20 Density ¶
    - depends_on definition 7.39 Darboux sums and the definite integral ¶
      - depends_on axiom 7.1 Completeness of $\R$ ¶
- depends_on theorem 7.7 Bolzano–Weierstrass ¶ ↺
- proves proof app:A-long-proofs@proof-101 ¶
proves proof app:A-long-proofs@proof-103 ¶

Supports

depends_on corollary A.181 The converse ¶

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:9138
`depends_on`	→	Helly's selection theorem	declared	appendices/A-long-proofs.tex:9138
`depends_on`	←	The converse	declared	appendices/A-long-proofs.tex:9380
`proves`	←	app:A-long-proofs@proof-103	declared	appendices/A-long-proofs.tex:9141