theorem 7.8 Cauchy criterion

open in the book · parts/02-mathematical-methods/05-real-analysis.tex:131 · p. 204

Rests on

depends_on corollary A.47 Cauchy completeness ¶
- 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 $\R$ ¶
      - depends_on definition A.36 Cut ¶ ↺
    - proves proof app:A-long-proofs@proof-29 ¶
  - 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 ¶

Supports

depends_on lemma A.288 The iteration converges ¶
- depends_on corollary A.289 The zero set is a graph ¶
- depends_on lemma A.290 $h$ is Lipschitz ¶
  - depends_on lemma A.291 $h$ is differentiable, with the stated derivative ¶
depends_on lemma A.454 Arzelà–Ascoli on an interval ¶
- depends_on theorem A.490 Equicontinuity of the normalised class ¶
- depends_on theorem A.470 $T$ is compact ¶
  - depends_on theorem A.471 Spectral decomposition and completeness in $L^{2}_{r}$ ¶
    - depends_on lemma A.472 The pairing identity ¶
- depends_on theorem A.455 Compact embedding of $W^{1,r}(a,b)$ into the continuous functions ¶
  - depends_on theorem A.456 Convexity in $q$ implies weak lower semicontinuity ¶
depends_on proposition 7.47 Comparison; absolute convergence ¶
- depends_on lemma 9.6 Weierstrass $M$-test; uniform limits are continuous ¶
  - depends_on proposition 9.139 Series for the complete integral of the first kind ¶
    - depends_on proposition 9.144 The pendulum equation at finite amplitude ¶
  - depends_on proposition 9.22 The exponential and its derivative ¶
    - depends_on theorem 9.23 Solution of a constant-coefficient system ¶
      - depends_on corollary 9.25 Normal modes of a diagonalizable system ¶
      - depends_on proposition 9.87 The distinguished limit, and which end carries the layer ¶
      - depends_on proposition 9.31 Lyapunov's first method: asymptotic stability ¶
      - depends_on proposition 9.43 The unmodulated line, and the feet of the tongues ¶
      - depends_on theorem 9.32 Hartman–Grobman; quoted ¶
      - depends_on theorem 9.26 Structure of the solutions ¶
  - depends_on theorem 9.8 Picard–Lindelöf ¶
    - depends_on corollary 32.5 Trajectories do not cross ¶
      - depends_on definition 32.29 Limit cycle ¶
      - depends_on definition 32.38 Poincaré section and return map ¶
      - depends_on theorem 32.30 Poincaré–Bendixson, restated from Part II ¶
    - depends_on corollary 9.9 Linear equations: existence on the whole interval ¶
      - depends_on definition 9.37 Fundamental matrix ¶
      - depends_on lemma A.439 Existence, uniqueness, and the structure of the zeros ¶
      - depends_on proposition 20.9 Linear resistance: exact motion, and the lost range ¶
      - depends_on proposition 9.103 The positive zeros and their spacing ¶
      - depends_on proposition 9.18 The linear equation: integrating factor ¶
      - depends_on theorem A.74 Flow of a time-dependent vector field ¶
      - depends_on theorem 9.23 Solution of a constant-coefficient system ¶ ↺
      - depends_on theorem 9.13 Dimension of the solution space ¶
    - depends_on proposition 32.4 The flow is a one-parameter group ¶
      - depends_on corollary 32.5 Trajectories do not cross ¶ ↺
    - depends_on proposition 10.16 Cauchy's characteristic strips ¶
    - depends_on proposition 20.7 Terminal speed and the approach to it ¶
      - depends_on example 20.8 A raindrop ¶
    - depends_on proposition 9.11 Continuous dependence on the initial data ¶
    - depends_on proposition 9.142 The differential equation of the sine amplitude ¶
    - depends_on theorem A.74 Flow of a time-dependent vector field ¶ ↺
    - depends_on theorem 13.125 Existence, uniqueness and smoothness of the flow ¶
      - depends_on lemma A.548 The joint flow is a translation action on the level set ¶
      - depends_on lemma A.567 The differential of the momentum map along the orbit ¶
      - depends_on proposition 13.131 Commuting fields have commuting flows ¶
      - depends_on proposition 13.132 Simultaneous straightening of commuting fields ¶
      - depends_on theorem 13.137 Commuting complete fields on a compact manifold ¶
    - depends_on theorem 9.34 Poincaré–Bendixson; quoted ¶
      - depends_on theorem 32.30 Poincaré–Bendixson, restated from Part II ¶ ↺
- depends_on proposition 7.48 Ratio test ¶
  - depends_on definition 7.53 $\ee$ ¶
    - depends_on lemma A.72 Iterated integral inequality ¶
      - depends_on theorem A.74 Flow of a time-dependent vector field ¶ ↺
    - depends_on lemma A.306 The standard smooth bump ¶
      - depends_on lemma A.307 Partition of unity on a compact manifold ¶
    - depends_on proposition 7.56 Irrationality ¶
      - depends_on remark 7.94 Three numbers, two kinds ¶
    - depends_on proposition 7.57 The limit definition ¶
      - depends_on lemma 7.67 The limit over real exponents ¶
      - depends_on remark 7.58 An identity, not a recipe ¶
  - depends_on definition 7.59 The real exponential ¶
    - depends_on lemma 7.60 Functional equation of the exponential ¶
      - depends_on proposition 7.68 The natural base ¶
      - depends_on proposition 7.61 The logarithm ¶
      - depends_on proposition 7.63 Laws of real powers ¶
  - depends_on definition 8.3 Exponential, sine, cosine ¶
    - depends_on lemma A.786 From the half-strip to the half plane ¶
      - depends_on corollary A.788 The two parameters are the same parameter ¶
      - depends_on lemma A.787 From the potential strip to the half plane ¶
    - depends_on lemma A.787 From the potential strip to the half plane ¶ ↺
    - depends_on lemma A.785 The hodograph image ¶
      - depends_on lemma A.786 From the half-strip to the half plane ¶ ↺
  - depends_on lemma 7.72 Addition theorems ¶
    - depends_on proposition 7.79 The polygon recursion ¶
      - depends_on corollary 7.81 The decimal expansion of $\pi$ ¶
      - depends_on proposition 7.83 Viète's product ¶
      - depends_on remark 7.82 What Archimedes proved ¶
      - depends_on remark 7.80 Why these are the polygon perimeters ¶
      - depends_on remark 7.84 One construction, two faces ¶
    - depends_on proposition 7.77 Special values, periodicity, and the kernel ¶
      - depends_on lemma 7.78 Chord, arc, tangent ¶
      - depends_on proposition 7.85 Irrationality of $\pi$ ¶
      - depends_on proposition 7.79 The polygon recursion ¶ ↺
      - depends_on proposition 7.83 Viète's product ¶ ↺
      - depends_on proposition 125.2 Crystallographic restriction ¶
      - depends_on remark 7.88 $\pi$ elsewhere in this treatise ¶
    - depends_on proposition 7.83 Viète's product ¶ ↺
    - depends_on proposition 125.2 Crystallographic restriction ¶ ↺
    - depends_on remark 7.88 $\pi$ elsewhere in this treatise ¶ ↺
    - depends_on remark 7.80 Why these are the polygon perimeters ¶ ↺
  - depends_on proposition 7.85 Irrationality of $\pi$ ¶ ↺
  - depends_on theorem 7.51 Termwise differentiation ¶
    - depends_on definition 7.74 $\pi$ ¶
      - depends_on lemma 7.75 The first quadrant ¶
      - depends_on proposition 7.76 $\pi$ as the circle constant ¶
      - depends_on proposition 7.79 The polygon recursion ¶ ↺
      - depends_on proposition 7.77 Special values, periodicity, and the kernel ¶ ↺
      - depends_on proposition 125.2 Crystallographic restriction ¶ ↺
    - depends_on lemma 7.93 A polynomial has at most $n$ roots ¶
    - depends_on lemma 7.71 Derivatives; the Pythagorean identity ¶
      - depends_on lemma 7.75 The first quadrant ¶ ↺
      - depends_on lemma 7.78 Chord, arc, tangent ¶ ↺
      - depends_on lemma 9.136 Wallis integrals ¶
      - depends_on proposition 7.76 $\pi$ as the circle constant ¶ ↺
      - depends_on proposition 7.85 Irrationality of $\pi$ ¶ ↺
      - depends_on proposition 125.2 Crystallographic restriction ¶ ↺
      - depends_on proposition 9.141 First properties ¶
      - depends_on remark 7.80 Why these are the polygon perimeters ¶ ↺
    - depends_on lemma 7.75 The first quadrant ¶ ↺
    - depends_on proposition 9.22 The exponential and its derivative ¶ ↺
- depends_on proposition 8.4 Euler's formula ¶
  - depends_on lemma A.786 From the half-strip to the half plane ¶ ↺
  - depends_on lemma 8.11 The fundamental $2\pi\ii$ ¶
    - depends_on definition 8.22 Isolated singularities; residue ¶
      - depends_on proposition 8.23 Residue at a simple pole ¶
      - depends_on proposition 17.96 Poles give asymptotics ¶
      - depends_on theorem 8.24 Residue theorem ¶
    - depends_on theorem 8.16 Cauchy integral formula ¶
      - depends_on theorem 8.17 Derivatives of all orders; Cauchy estimates ¶
      - depends_on theorem 8.21 Laurent expansion ¶
    - depends_on theorem 8.21 Laurent expansion ¶ ↺
    - depends_on theorem 8.24 Residue theorem ¶ ↺
  - depends_on lemma 17.14 Uniformly bounded sine sums ¶
    - depends_on lemma 17.15 A polynomial whose partial sum spikes at the origin ¶
      - depends_on proposition 17.16 An explicit continuous function with a divergent Fourier series ¶
- depends_on theorem 7.50 Power series; radius of convergence ¶
  - depends_on definition 7.59 The real exponential ¶ ↺
  - depends_on definition 8.3 Exponential, sine, cosine ¶ ↺
  - depends_on theorem 7.51 Termwise differentiation ¶ ↺
- depends_on theorem 7.51 Termwise differentiation ¶ ↺
depends_on proposition 7.85 Irrationality of $\pi$ ¶ ↺
depends_on proposition 7.48 Ratio test ¶ ↺
depends_on proposition A.519 The convolution exists ¶
- depends_on proposition A.524 Rate of decay ¶

Neighborhood

Every logical edge within two steps of this node.

declared and complete
partly declared
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Edges

type	direction	node	provenance	where
`depends_on`	→	Cauchy completeness	declared	parts/02-mathematical-methods/05-real-analysis.tex:135
`depends_on`	→	Convergence	declared	parts/02-mathematical-methods/05-real-analysis.tex:135
`depends_on`	←	The iteration converges	declared	appendices/A-long-proofs.tex:14481
`depends_on`	←	Arzelà–Ascoli on an interval	declared	appendices/A-long-proofs.tex:22311
`depends_on`	←	Comparison; absolute convergence	declared	parts/02-mathematical-methods/05-real-analysis.tex:914
`depends_on`	←	Irrationality of $\pi$	declared	parts/02-mathematical-methods/05-real-analysis.tex:2472
`depends_on`	←	Ratio test	declared	parts/02-mathematical-methods/05-real-analysis.tex:931
`depends_on`	←	The convolution exists	declared	appendices/A-long-proofs.tex:25355
`proves`	←	ch:05-real-analysis@proof-5	declared	parts/02-mathematical-methods/05-real-analysis.tex:138