proposition A.524 Rate of decay

Rests on

• depends_on proposition A.521 Every derivative passes onto $g$ ¶
  • depends_on lemma A.520 A $C^{1}$ limit ¶
    • depends_on definition 7.98 Functions of class $C^{1}$ ¶
      • 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 7.97 Partial derivative; gradient ¶
        • depends_on definition 7.26 Derivative of a function at a point ¶
        • depends_on definition 5.15 Basis ¶
    • 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 definition 7.39 Darboux sums and the definite integral ¶
        • depends_on theorem 7.24 Extreme value theorem ¶
        • depends_on theorem 7.25 Heine–Cantor: uniform continuity ¶
        • proves proof ch:05-real-analysis@proof-23 ¶
      • depends_on theorem 7.24 Extreme value theorem ¶ ↺
      • proves proof ch:05-real-analysis@proof-24 ¶
    • depends_on theorem 7.43 Fundamental theorem of calculus, II ¶
      • depends_on definition 7.41 Antiderivative ¶
        • depends_on corollary 7.36 cor:ana-mvt-consequences ¶
        • depends_on definition 7.26 Derivative of a function at a point ¶ ↺
      • depends_on theorem 7.42 Fundamental theorem of calculus, I ¶ ↺
      • proves proof ch:05-real-analysis@proof-25 ¶
    • proves proof app:A-long-proofs@proof-311 ¶
  • depends_on lemma A.518 The excised ball ¶
    • 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 example 7.130 The two Jacobians this treatise uses ¶
      • depends_on equation 7.131 eq:ana-nabla-sph-grad ¶
      • depends_on theorem 7.129 Change of variables in a multiple integral ¶
        • depends_on corollary 7.113 Inverse function theorem ¶
        • depends_on definition 7.125 Multiple integral ¶ ↺
        • depends_on remark 7.128 What the derivations below take as given ¶
        • proves proof ch:05-real-analysis@prooflink-3 ¶
    • depends_on theorem 7.129 Change of variables in a multiple integral ¶ ↺
    • proves proof app:A-long-proofs@proof-309 ¶
  • depends_on theorem 7.109 Leibniz integral rule ¶
    • depends_on definition 7.125 Multiple integral ¶ ↺
    • depends_on theorem 7.25 Heine–Cantor: uniform continuity ¶ ↺
    • depends_on theorem 7.35 Mean value theorem ¶
      • depends_on proposition 7.29 Linearity ¶
        • depends_on definition 7.11 Sum of functions ¶
        • depends_on equation 7.14 eq:ana-derivh ¶
        • depends_on proposition 7.6 Algebra of limits ¶
        • proves proof ch:05-real-analysis@proof-13 ¶
      • 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-19 ¶
    • proves proof ch:05-real-analysis@proof-67 ¶
  • proves proof app:A-long-proofs@proof-312 ¶
• depends_on proposition A.519 The convolution exists ¶
  • depends_on lemma A.518 The excised ball ¶ ↺
  • 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-33 ¶
      • 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 ¶
  • depends_on theorem 7.129 Change of variables in a multiple integral ¶ ↺
  • proves proof app:A-long-proofs@proof-310 ¶
• depends_on theorem 7.109 Leibniz integral rule ¶ ↺
• proves proof app:A-long-proofs@proof-315 ¶

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