lemma 9.6 Weierstrass $M$-test; uniform limits are continuous

open in the book · parts/02-mathematical-methods/07-odes-sturm-liouville.tex:192 · p. 274

Rests on

depends_on definition 7.20 Continuity at a point ¶
- depends_on definition 7.16 Limit ¶
  - depends_on definition 7.2 Absolute value ¶
  - depends_on definition 7.9 Real function ¶
    - depends_on definition 3.43 Map ¶
      - depends_on definition 3.24 Quantifiers ¶
      - depends_on definition 3.28 Set ¶
- depends_on equation 7.7 eq:ana-limit-left ¶
- depends_on equation 7.5 eq:ana-limit-right ¶
depends_on proposition 7.47 Comparison; absolute convergence ¶
- 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 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 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-34 ¶
  - depends_on definition 7.4 Convergence ¶ ↺
  - proves proof ch:05-real-analysis@proof-5 ¶
- proves proof ch:05-real-analysis@proof-28 ¶
proves proof ch:07-odes-sturm-liouville@proof-1 ¶

Supports

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 corollary 9.27 Linear stability ¶
      - depends_on corollary 9.41 Hill's equation ¶
      - depends_on proposition 9.31 Lyapunov's first method: asymptotic stability ¶
      - depends_on proposition 9.29 The planar classification ¶
      - depends_on theorem 9.39 Floquet ¶
    - depends_on proposition 9.29 The planar classification ¶ ↺
  - depends_on proposition 9.87 The distinguished limit, and which end carries the layer ¶
    - depends_on proposition 9.88 The model problem, and the error of the composite ¶
  - 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 32.15 Hartman–Grobman, restated from Part II ¶
      - depends_on definition 32.22 Bifurcation ¶
      - depends_on example 32.18 A linear centre that is really a stable focus ¶
  - depends_on theorem 9.26 Structure of the solutions ¶
    - depends_on corollary 9.27 Linear stability ¶ ↺
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 phenomenon 32.35 Self-sustained oscillation ¶
      - depends_on phenomenon 32.37 Frequency demultiplication and the first heard chaos ¶
  - depends_on definition 32.38 Poincaré section and return map ¶
    - depends_on definition 32.91 The circle map ¶
      - depends_on proposition 32.93 The circle map loses invertibility at $K=1$ ¶
    - depends_on example 32.70 The Hénon map ¶
    - depends_on proposition 32.93 The circle map loses invertibility at $K=1$ ¶ ↺
    - depends_on proposition 32.95 Intermittency: the scaling of the laminar phase ¶
    - depends_on proposition 32.39 Stability of a periodic orbit ¶
      - depends_on phenomenon 32.87 Universal period doubling ¶
      - depends_on proposition 32.84 Fixed points and the first period doubling ¶
  - depends_on theorem 32.30 Poincaré–Bendixson, restated from Part II ¶
    - depends_on corollary 32.32 No chaos in a planar flow ¶
- depends_on corollary 9.9 Linear equations: existence on the whole interval ¶
  - depends_on definition 9.37 Fundamental matrix ¶
    - depends_on proposition 9.38 Liouville's formula ¶
      - depends_on corollary 9.41 Hill's equation ¶ ↺
      - depends_on theorem 9.39 Floquet ¶ ↺
    - depends_on theorem 9.39 Floquet ¶ ↺
  - depends_on lemma A.439 Existence, uniqueness, and the structure of the zeros ¶
    - depends_on lemma A.442 The Wronskian of the accessory equation is constant ¶
      - depends_on definition A.467 Green function of the shifted problem ¶
      - depends_on theorem A.443 Sturm separation theorem ¶
    - depends_on theorem A.441 Continuous dependence on the data and on the initial point ¶
      - depends_on corollary A.445 Displacing the initial point ¶
  - depends_on proposition 20.9 Linear resistance: exact motion, and the lost range ¶
    - depends_on remark 20.11 What has no closed form, and what to do about it ¶
  - depends_on proposition 9.103 The positive zeros and their spacing ¶
  - depends_on proposition 9.18 The linear equation: integrating factor ¶
    - depends_on proposition 9.87 The distinguished limit, and which end carries the layer ¶ ↺
  - 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 9.23 Solution of a constant-coefficient system ¶ ↺
  - depends_on theorem 9.13 Dimension of the solution space ¶
    - depends_on definition 9.37 Fundamental matrix ¶ ↺
    - depends_on proposition 28.18 The three regimes ¶
    - depends_on proposition 28.4 General solution and isochronism ¶
      - depends_on phenomenon 28.9 Isochronism of the small-amplitude pendulum ¶
    - depends_on proposition 9.106 The modified series and what it does not do ¶
    - depends_on proposition 9.28 The scalar constant-coefficient equation ¶
      - depends_on proposition 9.88 The model problem, and the error of the composite ¶ ↺
    - depends_on proposition 9.16 Wronskian test for independence ¶
- 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 theorem 9.34 Poincaré–Bendixson; quoted ¶
  - depends_on theorem 32.30 Poincaré–Bendixson, restated from Part II ¶ ↺

Neighborhood

Every logical edge within two steps of this node.

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partly declared
a check failed
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declared in the source
inferred from structure

Edges

type	direction	node	provenance	where
`depends_on`	→	Continuity at a point	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:207
`depends_on`	→	Comparison; absolute convergence	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:207
`depends_on`	←	Series for the complete integral of the first kind	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:5280
`depends_on`	←	The exponential and its derivative	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:753
`depends_on`	←	Picard–Lindelöf	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:271
`proves`	←	ch:07-odes-sturm-liouville@proof-1	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:210