theorem A.173 Helly's selection theorem
open in the book ·
appendices/A-long-proofs.tex:9033
· p. 2876
Rests on
-
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
definition 7.39
Darboux sums and the definite integral
¶
-
depends_on
definition 11.20
Density
¶
-
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 ¶
-
depends_on
corollary A.45
Archimedean property and density of $\Q$
¶
- proves proof app:A-long-proofs@proof-101 ¶
Supports
-
depends_on
lemma A.176
Helly–Bray
¶
- depends_on corollary A.181 The converse ¶
- depends_on lemma A.175 Tightness makes the Helly limit proper ¶
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 |
→ | Distribution function | declared | appendices/A-long-proofs.tex:9038 |
depends_on |
→ | Bolzano–Weierstrass | declared | appendices/A-long-proofs.tex:9038 |
depends_on |
← | Helly–Bray | declared | appendices/A-long-proofs.tex:9138 |
depends_on |
← | Tightness makes the Helly limit proper | declared | appendices/A-long-proofs.tex:9114 |
proves |
← | app:A-long-proofs@proof-101 | declared | appendices/A-long-proofs.tex:9045 |