lemma A.454 Arzelà–Ascoli on an interval

open in the book · appendices/A-long-proofs.tex:22297 · p. 3015

Rests on

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 $\R$ ¶
      - depends_on definition A.36 Cut ¶ ↺
    - 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 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 ¶
  - depends_on definition 7.2 Absolute value ¶
- proves proof ch:05-real-analysis@proof-5 ¶
proves proof app:A-long-proofs@proof-270 ¶

Supports

depends_on theorem A.490 Equicontinuity of the normalised class ¶
depends_on theorem A.470 $T$ is compact ¶
- depends_on theorem A.471 Spectral decomposition and completeness in $L^{2}_{r}$ ¶
  - depends_on lemma A.472 The pairing identity ¶
depends_on theorem A.455 Compact embedding of $W^{1,r}(a,b)$ into the continuous functions ¶
- depends_on theorem A.456 Convexity in $q$ implies weak lower semicontinuity ¶

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`	→	Bolzano–Weierstrass	declared	appendices/A-long-proofs.tex:22311
`depends_on`	→	Cauchy criterion	declared	appendices/A-long-proofs.tex:22311
`depends_on`	←	Equicontinuity of the normalised class	declared	appendices/A-long-proofs.tex:24032
`depends_on`	←	$T$ is compact	declared	appendices/A-long-proofs.tex:23151
`depends_on`	←	Compact embedding of $W^{1,r}(a,b)$ into the continuous functions	declared	appendices/A-long-proofs.tex:22385
`proves`	←	app:A-long-proofs@proof-270	declared	appendices/A-long-proofs.tex:22314