remark 7.95 Where $\varphi$ is used

open in the book · parts/02-mathematical-methods/05-real-analysis.tex:2854 · p. 241

remark -- no derivation owed by its kind

Rests on

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 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 definition 7.45 Series ¶ ↺
    - proves proof ch:05-real-analysis@proof-27 ¶
  - 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 7.92 Continued fraction and the Fibonacci ratios ¶
- depends_on proposition 7.91 The golden ratio ¶ ↺
- depends_on proposition 7.6 Algebra of limits ¶ ↺
- proves proof ch:05-real-analysis@proof-60 ¶

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type	direction	node	provenance	where
`depends_on`	→	The golden ratio	declared	parts/02-mathematical-methods/05-real-analysis.tex:2876
`depends_on`	→	Continued fraction and the Fibonacci ratios	declared	parts/02-mathematical-methods/05-real-analysis.tex:2876