remark 7.87 Priority, and what irrationality does not give

Rests on

• depends_on proposition 7.85 Irrationality of $\pi$ ¶
  • depends_on lemma 7.86 Terminating or repeating decimals are rational ¶
    • 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 ¶ ↺
    • depends_on proposition 7.46 Geometric series ¶
      • 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 definition 7.45 Series ¶ ↺
      • proves proof ch:05-real-analysis@proof-27 ¶
    • proves proof ch:05-real-analysis@proof-57 ¶
  • 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 ¶
        • depends_on definition 7.4 Convergence ¶ ↺
        • depends_on proposition 7.3 Triangle inequality ¶
        • proves proof ch:05-real-analysis@proof-3 ¶
      • 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 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 7.27 Differentiable implies continuous ¶
        • depends_on definition 7.20 Continuity at a point ¶
        • depends_on definition 7.26 Derivative of a function at a point ¶
        • depends_on proposition 7.6 Algebra of limits ¶ ↺
        • proves proof ch:05-real-analysis@proof-11 ¶
      • depends_on proposition 7.48 Ratio test ¶
        • depends_on proposition 7.47 Comparison; absolute convergence ¶ ↺
        • depends_on proposition 7.46 Geometric series ¶ ↺
        • depends_on theorem 7.8 Cauchy criterion ¶ ↺
        • proves proof ch:05-real-analysis@proof-29 ¶
      • 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 axiom 7.1 Completeness of $\R$ ¶ ↺
        • depends_on definition 7.20 Continuity at a point ¶ ↺
        • proves proof ch:05-real-analysis@proof-8 ¶
      • 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 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 7.75 The first quadrant ¶
      • depends_on corollary 7.36 cor:ana-mvt-consequences ¶
        • depends_on theorem 7.35 Mean value theorem ¶
        • proves proof ch:05-real-analysis@proof-20 ¶
      • 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 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 ¶

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