theorem A.490 Equicontinuity of the normalised class

Rests on

• depends_on lemma A.488 Small chords cut off small arcs ¶
  • 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 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 ¶
        • 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 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 3.38 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 ¶
• depends_on lemma A.489 Courant–Lebesgue ¶
  • depends_on definition A.476 Disc-type surfaces and the two functionals ¶
    • depends_on definition 13.22 First fundamental coefficients ¶
      • depends_on definition 13.21 First fundamental form ¶
        • depends_on definition 13.11 Arc length ¶
        • depends_on definition 13.18 Surface ¶
      • depends_on definition 13.19 Parameter curves ¶
        • depends_on definition 13.8 Curve ¶
        • depends_on definition 13.18 Surface ¶ ↺
    • depends_on equation 16.60 eq:calcvar-dirichlet-integral ¶
  • depends_on lemma A.452 Young and Hölder ¶
    • depends_on definition A.450 The space $W^{1,r}(a,b)$ ¶
      • depends_on definition 10.85 Sobolev space ¶
        • depends_on definition 12.2 Hilbert space ¶
        • depends_on definition 10.83 Weak solution ¶
      • depends_on theorem 7.42 Fundamental theorem of calculus, I ¶
        • depends_on definition 7.26 Derivative of a function at a point ¶
        • depends_on theorem 7.40 Continuous functions are integrable ¶
        • depends_on theorem 7.24 Extreme value theorem ¶
        • proves proof ch:05-real-analysis@proof-24 ¶
    • depends_on theorem 7.38 Taylor's theorem with Lagrange remainder ¶
      • depends_on proposition 7.30 Leibniz rule ¶
        • depends_on definition 7.12 Product of functions ¶
        • depends_on equation 7.14 eq:ana-derivh ¶
        • depends_on proposition 7.27 Differentiable implies continuous ¶
        • depends_on proposition 7.6 Algebra of limits ¶
        • proves proof ch:05-real-analysis@proof-14 ¶
      • depends_on theorem 7.34 Rolle ¶
        • depends_on lemma 7.33 Fermat: interior extremum ¶
        • depends_on theorem 7.24 Extreme value theorem ¶ ↺
        • proves proof ch:05-real-analysis@proof-18 ¶
      • proves proof ch:05-real-analysis@proof-22 ¶
    • proves proof app:A-long-proofs@proof-268 ¶
  • proves proof app:A-long-proofs@proof-290 ¶
• depends_on lemma A.454 Arzelà–Ascoli on an interval ¶
  • depends_on theorem 7.7 Bolzano–Weierstrass ¶ ↺
  • 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 ¶ ↺
    • proves proof ch:05-real-analysis@proof-5 ¶
  • proves proof app:A-long-proofs@proof-270 ¶
• proves proof app:A-long-proofs@proof-291 ¶

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