proposition 7.3 Triangle inequality

open in the book · parts/02-mathematical-methods/05-real-analysis.tex:48 · p. 203

Rests on

depends_on definition 7.2 Absolute value ¶
proves proof ch:05-real-analysis@proof-1 ¶

Supports

depends_on proposition 7.6 Algebra of limits ¶
- depends_on lemma 7.67 The limit over real exponents ¶
  - depends_on proposition 7.68 The natural base ¶
    - depends_on remark 7.70 What the condition says ¶
- 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 remark 7.88 $\pi$ elsewhere in this treatise ¶
    - depends_on proposition 7.83 Viète's product ¶
      - depends_on remark 7.84 One construction, two faces ¶
    - 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 7.79 The polygon recursion ¶ ↺
    - depends_on proposition 7.85 Irrationality of $\pi$ ¶
      - depends_on remark 7.94 Three numbers, two kinds ¶
      - depends_on remark 7.87 Priority, and what irrationality does not give ¶
    - 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 125.3 What the theorem does and does not forbid ¶
    - 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 7.49 Cauchy product ¶
  - 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 corollary 7.69 $\ee$ is the unique self-reproducing base ¶
      - depends_on definition 7.62 Real powers ¶
      - depends_on lemma 7.67 The limit over real exponents ¶ ↺
      - depends_on proposition 7.68 The natural base ¶ ↺
      - depends_on proposition 7.63 Laws of real powers ¶
    - depends_on proposition 7.63 Laws of real powers ¶ ↺
  - depends_on lemma 7.71 Derivatives; the Pythagorean identity ¶
    - depends_on lemma 7.75 The first quadrant ¶
      - depends_on proposition 7.76 $\pi$ as the circle constant ¶
      - depends_on proposition 7.77 Special values, periodicity, and the kernel ¶ ↺
      - depends_on proposition 125.2 Crystallographic restriction ¶ ↺
    - depends_on lemma 7.78 Chord, arc, tangent ¶ ↺
    - depends_on lemma 9.136 Wallis integrals ¶
      - depends_on proposition 9.139 Series for the complete integral of the first kind ¶
    - 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 proposition 9.143 A cubic quadrature between two turning points ¶
      - depends_on proposition 9.142 The differential equation of the sine amplitude ¶
    - depends_on remark 7.80 Why these are the polygon perimeters ¶ ↺
  - depends_on lemma 7.72 Addition theorems ¶ ↺
- depends_on proposition 7.31 Chain rule ¶
  - depends_on corollary 7.44 Substitution and integration by parts ¶
    - depends_on definition 8.8 Contour integral ¶
    - depends_on lemma 16.20 Mixed form ¶
      - depends_on theorem 16.35 System of Euler–Lagrange equations ¶
      - depends_on theorem 16.22 Euler–Lagrange ¶
      - depends_on theorem 16.43 Euler's rule for integral constraints ¶
    - depends_on lemma 9.136 Wallis integrals ¶ ↺
    - depends_on proposition 7.145 Delta as a limit; elementary properties ¶
    - depends_on proposition 16.50 The second variation ¶
    - depends_on proposition 5.95 The focal polar equation of a conic ¶
    - depends_on proposition 9.143 A cubic quadrature between two turning points ¶ ↺
    - depends_on proposition 9.144 The pendulum equation at finite amplitude ¶
    - depends_on theorem 16.38 Euler–Poisson equation ¶
      - depends_on theorem 30.56 The beam equation ¶
      - depends_on theorem 30.64 The Kirchhoff plate equation ¶
  - depends_on lemma A.479 Conformal invariance of the Dirichlet integral ¶
    - depends_on lemma A.487 The normalisation costs nothing ¶
      - depends_on theorem A.492 Douglas' conformality theorem; quoted ¶
    - depends_on lemma A.485 Conformal automorphisms of the disc ¶
      - depends_on definition A.486 The normalised admissible class ¶
      - depends_on lemma A.487 The normalisation costs nothing ¶ ↺
  - depends_on lemma 27.15 Velocity and acceleration in plane polar coordinates ¶
  - depends_on proposition 7.111 Jacobi's formula, cofactor form ¶
  - depends_on proposition 7.63 Laws of real powers ¶ ↺
  - depends_on proposition 16.26 Covariance under change of dependent variable ¶
  - depends_on proposition 10.15 Method of characteristics, first order ¶
    - depends_on definition 10.93 Scalar conservation law ¶
      - depends_on definition A.150 Kruzhkov entropy pair ¶
      - depends_on definition 10.95 Weak solution of a conservation law ¶
      - depends_on example 10.97 Burgers shock ¶
      - depends_on proposition 10.94 Gradient catastrophe ¶
    - depends_on proposition 10.16 Cauchy's characteristic strips ¶
    - depends_on proposition 10.94 Gradient catastrophe ¶ ↺
    - depends_on remark 31.90 Why a compression wave must steepen ¶
    - depends_on remark 23.3 What Part II owes this chapter ¶
  - depends_on theorem 16.29 Beltrami identity ¶
    - depends_on proposition 16.105 The optical invariant in a stratified medium ¶
    - depends_on proposition 16.39 Ostrogradsky instability ¶
  - depends_on theorem 10.8 The type is a coordinate invariant ¶
    - depends_on lemma A.90 Affine normalization ¶
      - depends_on lemma A.91 Smallness after rescaling ¶
    - depends_on proposition 10.13 Canonical form in two variables ¶
      - depends_on theorem A.83 Local canonical form for an elliptic operator with Hölder coefficients ¶
      - depends_on theorem 10.55 d'Alembert's formula ¶
    - depends_on proposition 10.11 What a characteristic surface is ¶
      - depends_on corollary 10.12 Characteristics of the three canonical operators ¶
      - depends_on definition A.102 Normal Cauchy problem of order $k$ ¶
      - depends_on theorem 10.22 Cauchy–Kovalevskaya ¶
- depends_on proposition 7.32 Derivative of the inverse function ¶
  - depends_on definition 9.140 Amplitude and the Jacobi elliptic functions ¶
    - depends_on proposition 9.143 A cubic quadrature between two turning points ¶ ↺
    - depends_on proposition 9.141 First properties ¶ ↺
    - depends_on proposition 9.144 The pendulum equation at finite amplitude ¶ ↺
  - depends_on proposition 7.61 The logarithm ¶ ↺
  - depends_on proposition 9.141 First properties ¶ ↺
  - depends_on proposition 9.19 Separation of variables ¶
- depends_on proposition 7.29 Linearity ¶
  - depends_on theorem 7.35 Mean value theorem ¶
    - depends_on corollary 7.36 cor:ana-mvt-consequences ¶
      - depends_on definition 7.41 Antiderivative ¶
      - depends_on lemma 7.67 The limit over real exponents ¶ ↺
      - depends_on lemma 7.75 The first quadrant ¶ ↺
      - depends_on lemma 7.78 Chord, arc, tangent ¶ ↺
      - depends_on lemma 9.10 Grönwall's inequality ¶
      - depends_on proposition 7.108 Euler's theorem on homogeneous functions ¶
      - depends_on proposition 7.61 The logarithm ¶ ↺
      - depends_on proposition 9.19 Separation of variables ¶ ↺
      - depends_on theorem 7.115 Constant rank ¶
    - depends_on lemma 7.93 A polynomial has at most $n$ roots ¶
    - depends_on lemma A.500 Graphs and $C^{1}$ images have zero content ¶
      - depends_on corollary A.514 Degeneracy on a negligible set ¶
      - depends_on lemma A.512 The boundary strip is thin ¶
    - depends_on lemma A.73 Differentiation under the integral sign ¶
      - depends_on lemma A.77 Poincaré lemma, converse form ¶
    - depends_on lemma A.287 The Newton map contracts ¶
      - depends_on lemma A.288 The iteration converges ¶
    - depends_on lemma A.290 $h$ is Lipschitz ¶
      - depends_on lemma A.291 $h$ is differentiable, with the stated derivative ¶
    - depends_on lemma A.440 Grönwall's inequality ¶
      - depends_on theorem A.441 Continuous dependence on the data and on the initial point ¶
    - depends_on lemma A.226 The level factor ¶
    - depends_on lemma A.468 The kernel is well defined, symmetric and Lipschitz ¶
      - depends_on theorem A.470 $T$ is compact ¶
    - depends_on lemma 11.68 Jensen's inequality for the logarithm ¶
    - depends_on proposition 7.105 Clairaut–Schwarz ¶
      - depends_on definition 10.4 The second-order operator ¶
      - depends_on lemma A.75 Differentiating a pullback along a flow ¶
      - depends_on lemma 22.20 The symplectic condition ¶
      - depends_on proposition 7.123 Second-order identities of the nabla calculus ¶
      - depends_on proposition 30.25 Twenty-one constants ¶
      - depends_on proposition 30.14 Saint-Venant compatibility is necessary ¶
      - depends_on proposition 22.30 Properties of the Poisson bracket ¶
      - depends_on proposition 13.148 prop:mfd-torsion-tensor ¶
      - depends_on proposition 10.16 Cauchy's characteristic strips ¶ ↺
      - depends_on theorem 7.132 Stokes ¶
      - depends_on theorem 7.106 Taylor's theorem in several variables ¶
      - depends_on theorem 13.152 Riemann tensor; Ricci identity with torsion ¶
      - … 2 more
    - depends_on theorem 7.100 $C^{1}$ implies differentiable ¶
      - depends_on lemma A.291 $h$ is differentiable, with the stated derivative ¶ ↺
      - depends_on remark 7.101 Partial derivatives alone do not suffice ¶
      - depends_on theorem 7.112 Implicit function theorem ¶
      - depends_on theorem A.286 Implicit function theorem ¶
    - … 2 more
- depends_on proposition 7.27 Differentiable implies continuous ¶
  - depends_on proposition 7.31 Chain rule ¶ ↺
  - depends_on proposition 7.68 The natural base ¶ ↺
  - depends_on proposition 7.61 The logarithm ¶ ↺
  - depends_on proposition 7.30 Leibniz rule ¶
    - depends_on corollary 7.44 Substitution and integration by parts ¶ ↺
    - depends_on proposition 7.108 Euler's theorem on homogeneous functions ¶ ↺
    - depends_on proposition 7.111 Jacobi's formula, cofactor form ¶ ↺
    - depends_on proposition 7.122 Product identities of the nabla calculus ¶
    - depends_on proposition 5.2 Jacobi's formula, column form ¶
      - depends_on lemma 44.10 Divergence theorem on $(M,g)$ ¶
      - depends_on lemma 44.7 Variation of inverse and determinant ¶
      - depends_on proposition 7.111 Jacobi's formula, cofactor form ¶ ↺
    - 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 theorem 7.38 Taylor's theorem with Lagrange remainder ¶
      - depends_on definition 32.57 The Hénon–Heiles Hamiltonian ¶
      - depends_on definition 32.12 Linearization ¶
      - depends_on lemma A.754 Exact depth expansion ¶
      - depends_on lemma A.187 Exponentiation ¶
      - depends_on lemma A.138 The flat exponential ¶
      - depends_on lemma A.452 Young and Hölder ¶
      - depends_on lemma A.212 Second-order flatness ¶
      - depends_on lemma A.209 Uniform third-order remainder ¶
      - depends_on phenomenon 28.13 Universality of small oscillations ¶
      - depends_on proposition 16.50 The second variation ¶ ↺
      - depends_on proposition 17.67 Holomorphy ¶
      - depends_on proposition 9.88 The model problem, and the error of the composite ¶ ↺
      - … 2 more
  - 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 proposition 9.22 The exponential and its derivative ¶
      - depends_on theorem 9.23 Solution of a constant-coefficient system ¶
- 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 remark 7.70 What the condition says ¶ ↺
- depends_on proposition 7.68 The natural base ¶ ↺
- depends_on proposition 7.92 Continued fraction and the Fibonacci ratios ¶
  - depends_on remark 7.95 Where $\varphi$ is used ¶
- depends_on proposition 7.30 Leibniz rule ¶ ↺
- depends_on proposition 7.83 Viète's product ¶ ↺
- … 1 more
depends_on proposition 7.5 Uniqueness of the limit ¶

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Edges

type	direction	node	provenance	where
`depends_on`	→	Absolute value	declared	parts/02-mathematical-methods/05-real-analysis.tex:50
`depends_on`	←	Algebra of limits	declared	parts/02-mathematical-methods/05-real-analysis.tex:87
`depends_on`	←	Uniqueness of the limit	declared	parts/02-mathematical-methods/05-real-analysis.tex:73
`proves`	←	ch:05-real-analysis@proof-1	declared	parts/02-mathematical-methods/05-real-analysis.tex:53