proposition 7.143 The Hausdorff dimension never exceeds the box dimension

open in the book · parts/02-mathematical-methods/05-real-analysis.tex:4874 · p. 262

Rests on

depends_on definition 7.142 Box-counting dimension ¶
- depends_on definition 7.140 Hausdorff measure ¶
  - depends_on axiom 7.1 Completeness of $\R$ ¶
  - depends_on definition 7.138 Set of measure zero ¶
    - depends_on definition 7.125 Multiple integral ¶
      - depends_on axiom 7.1 Completeness of $\R$ ¶ ↺
      - depends_on definition 7.39 Darboux sums and the definite integral ¶
    - depends_on definition 3.65 Countable set ¶
      - depends_on axiom 3.33 Peano axioms ¶
      - depends_on definition 3.62 Equipotence ¶
  - 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 6.9 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 lemma 7.139 A countable union of null sets is null ¶
- depends_on definition 7.138 Set of measure zero ¶ ↺
- 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 proposition 3.67 Countability of the plane of naturals ¶
  - depends_on definition 3.65 Countable set ¶ ↺
  - depends_on definition 3.47 Bijective map ¶
  - depends_on definition 3.38 Ordered pair and Cartesian product ¶ ↺
  - proves proof ch:01-logic-sets@proof-13 ¶
- proves proof ch:05-real-analysis@proof-81 ¶
depends_on proposition 7.141 The critical exponent, and the Hausdorff dimension ¶
- depends_on definition 7.140 Hausdorff measure ¶ ↺
- depends_on definition 7.138 Set of measure zero ¶ ↺
- depends_on lemma 7.139 A countable union of null sets is null ¶ ↺
- proves proof ch:05-real-analysis@proof-82 ¶
proves proof ch:05-real-analysis@proof-83 ¶

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type	direction	node	provenance	where
`depends_on`	→	Box-counting dimension	declared	parts/02-mathematical-methods/05-real-analysis.tex:4883
`depends_on`	→	A countable union of null sets is null	declared	parts/02-mathematical-methods/05-real-analysis.tex:4883
`depends_on`	→	The critical exponent, and the Hausdorff dimension	declared	parts/02-mathematical-methods/05-real-analysis.tex:4883
`proves`	←	ch:05-real-analysis@proof-83	declared	parts/02-mathematical-methods/05-real-analysis.tex:4887