remark 7.94 Three numbers, two kinds

Rests on

• depends_on corollary 3.68 The rationals are countable ¶
  • depends_on definition 3.65 Countable set ¶
    • depends_on axiom 3.33 Peano axioms ¶
    • depends_on definition 3.62 Equipotence ¶
      • depends_on definition 3.47 Bijective map ¶
      • depends_on definition 3.40 Cardinal ¶
  • depends_on equation 3.76 eq:inf-Z-bijection ¶
  • 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:01-logic-sets@proof-14 ¶
• depends_on lemma 7.139 A countable union of null sets is null ¶
  • 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 axiom 7.1 Completeness of $\R$ ¶ ↺
    • depends_on definition 3.65 Countable set ¶ ↺
  • 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 ¶ ↺
  • proves proof ch:05-real-analysis@proof-81 ¶
• depends_on proposition 7.56 Irrationality ¶
  • depends_on definition 7.53 $\ee$ ¶
    • depends_on definition 7.45 Series ¶ ↺
    • depends_on proposition 7.48 Ratio test ¶
      • depends_on proposition 7.47 Comparison; absolute convergence ¶
        • depends_on corollary A.46 Monotone convergence ¶ ↺
        • depends_on definition 7.45 Series ¶ ↺
        • depends_on theorem 7.8 Cauchy criterion ¶
        • proves proof ch:05-real-analysis@proof-28 ¶
      • depends_on proposition 7.46 Geometric series ¶ ↺
      • depends_on theorem 7.8 Cauchy criterion ¶ ↺
      • proves proof ch:05-real-analysis@proof-29 ¶
  • depends_on lemma 7.54 Truncation error ¶
    • depends_on equation 7.31 eq:ana-e-series ¶
    • depends_on proposition 7.46 Geometric series ¶ ↺
    • proves proof ch:05-real-analysis@proof-34 ¶
  • proves proof ch:05-real-analysis@proof-36 ¶
• depends_on proposition 7.91 The golden ratio ¶
  • depends_on definition 7.90 The golden ratio ¶
    • depends_on definition 7.62 Real powers ¶
      • depends_on proposition 7.61 The logarithm ¶
        • depends_on corollary 7.36 cor:ana-mvt-consequences ¶
        • depends_on corollary A.46 Monotone convergence ¶ ↺
        • depends_on equation 7.42 eq:ana-exp-derivative ¶
        • depends_on lemma 7.60 Functional equation of the exponential ¶
        • depends_on proposition 7.32 Derivative of the inverse function ¶
        • depends_on proposition 7.27 Differentiable implies continuous ¶
        • depends_on theorem 7.23 Intermediate value theorem ¶
        • proves proof ch:05-real-analysis@proof-39 ¶
    • depends_on proposition 7.63 Laws of real powers ¶
      • depends_on definition 7.62 Real powers ¶ ↺
      • depends_on equation 7.42 eq:ana-exp-derivative ¶ ↺
      • depends_on lemma 7.60 Functional equation of the exponential ¶ ↺
      • depends_on proposition 7.31 Chain rule ¶
        • depends_on definition 7.13 Composite function ¶
        • depends_on proposition 7.27 Differentiable implies continuous ¶ ↺
        • depends_on proposition 7.6 Algebra of limits ¶
        • proves proof ch:05-real-analysis@proof-15 ¶
      • depends_on proposition 7.61 The logarithm ¶ ↺
      • proves proof ch:05-real-analysis@proof-40 ¶
  • depends_on lemma 7.86 Terminating or repeating decimals are rational ¶
    • depends_on definition 7.45 Series ¶ ↺
    • depends_on proposition 7.46 Geometric series ¶ ↺
    • proves proof ch:05-real-analysis@proof-57 ¶
  • depends_on proposition 7.63 Laws of real powers ¶ ↺
  • depends_on proposition 7.89 Irrationality of square roots ¶
    • depends_on definition 7.62 Real powers ¶ ↺
    • depends_on lemma 7.65 The integer part ¶
      • depends_on axiom 3.33 Peano axioms ¶ ↺
      • 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 lemma 7.64 Well-ordering of $\N$ ¶
        • depends_on axiom 3.33 Peano axioms ¶ ↺
        • depends_on axiom 3.34 Principle of induction ¶
        • proves proof ch:05-real-analysis@proof-41 ¶
      • proves proof ch:05-real-analysis@proof-42 ¶
    • depends_on lemma 7.64 Well-ordering of $\N$ ¶ ↺
    • depends_on proposition 7.63 Laws of real powers ¶ ↺
    • proves proof ch:05-real-analysis@proof-58 ¶
  • proves proof ch:05-real-analysis@proof-59 ¶
• depends_on proposition 7.85 Irrationality of $\pi$ ¶
  • depends_on lemma 7.86 Terminating or repeating decimals are rational ¶ ↺
  • depends_on lemma 7.71 Derivatives; the Pythagorean identity ¶
    • depends_on equation 7.53 eq:ana-cos-sin-series ¶
    • depends_on equation 5.3 eq:lin-binomial ¶
    • depends_on proposition 7.49 Cauchy product ¶
      • depends_on corollary A.46 Monotone convergence ¶ ↺
      • depends_on definition 7.45 Series ¶ ↺
      • depends_on proposition 7.6 Algebra of limits ¶ ↺
      • proves proof ch:05-real-analysis@proof-30 ¶
    • depends_on theorem 7.51 Termwise differentiation ¶
      • depends_on proposition 7.47 Comparison; absolute convergence ¶ ↺
      • depends_on proposition 7.27 Differentiable implies continuous ¶ ↺
      • depends_on proposition 7.48 Ratio test ¶ ↺
      • depends_on proposition 7.6 Algebra of limits ¶ ↺
      • depends_on theorem 7.50 Power series; radius of convergence ¶
        • depends_on axiom 7.1 Completeness of $\R$ ¶ ↺
        • depends_on proposition 7.47 Comparison; absolute convergence ¶ ↺
        • depends_on proposition 7.46 Geometric series ¶ ↺
        • proves proof ch:05-real-analysis@proof-31 ¶
      • proves proof ch:05-real-analysis@proof-32 ¶
    • proves proof ch:05-real-analysis@proof-46 ¶
  • depends_on proposition 7.77 Special values, periodicity, and the kernel ¶
    • depends_on definition 7.74 $\pi$ ¶
      • depends_on axiom 7.1 Completeness of $\R$ ¶ ↺
      • depends_on lemma 7.73 $\cos 2 < 0$ ¶
        • depends_on equation 7.53 eq:ana-cos-sin-series ¶ ↺
        • depends_on proposition 7.52 Alternating series test ¶
        • proves proof ch:05-real-analysis@proof-48 ¶
      • depends_on theorem 7.23 Intermediate value theorem ¶ ↺
      • depends_on theorem 7.51 Termwise differentiation ¶ ↺
    • depends_on equation 7.58 eq:ana-half-angle ¶
    • depends_on lemma 7.65 The integer part ¶ ↺
    • depends_on lemma 7.75 The first quadrant ¶
      • depends_on corollary 7.36 cor:ana-mvt-consequences ¶ ↺
      • depends_on definition 7.74 $\pi$ ¶ ↺
      • depends_on lemma 7.71 Derivatives; the Pythagorean identity ¶ ↺
      • depends_on theorem 7.23 Intermediate value theorem ¶ ↺
      • depends_on theorem 7.51 Termwise differentiation ¶ ↺
      • proves proof ch:05-real-analysis@proof-49 ¶
    • depends_on lemma 7.72 Addition theorems ¶
      • depends_on equation 7.53 eq:ana-cos-sin-series ¶ ↺
      • depends_on equation 5.3 eq:lin-binomial ¶ ↺
      • depends_on proposition 7.49 Cauchy product ¶ ↺
      • depends_on proposition 7.48 Ratio test ¶ ↺
      • depends_on proposition 7.6 Algebra of limits ¶ ↺
      • proves proof ch:05-real-analysis@proof-47 ¶
    • proves proof ch:05-real-analysis@proof-51 ¶
  • depends_on proposition 7.48 Ratio test ¶ ↺
  • depends_on theorem 7.8 Cauchy criterion ¶ ↺
  • 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 definition 7.16 Limit ¶
        • depends_on definition 7.9 Real function ¶
    • 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 ¶
    • proves proof ch:05-real-analysis@proof-25 ¶
  • proves proof ch:05-real-analysis@proof-56 ¶
• depends_on proposition 3.67 Countability of the plane of naturals ¶ ↺
• depends_on theorem 3.69 Uncountability of the continuum ¶
  • depends_on definition 3.65 Countable set ¶ ↺
  • depends_on definition 3.46 Surjective map ¶
  • proves proof ch:01-logic-sets@proof-15 ¶

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