theorem A.40 Least-upper-bound property

open in the book · appendices/A-long-proofs.tex:2959 · p. 2810

Rests on

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 ¶

Supports

depends_on corollary A.45 Archimedean property and density of $\Q$ ¶
- depends_on lemma 7.65 The integer part ¶
  - 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 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 remark 7.84 One construction, two faces ¶
    - 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 7.89 Irrationality of square roots ¶
    - depends_on proposition 7.91 The golden ratio ¶
      - depends_on proposition 7.92 Continued fraction and the Fibonacci ratios ¶
      - depends_on proposition 125.2 Crystallographic restriction ¶ ↺
      - depends_on remark 7.95 Where $\varphi$ is used ¶
      - depends_on remark 7.94 Three numbers, two kinds ¶ ↺
- depends_on theorem 7.7 Bolzano–Weierstrass ¶
  - 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 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 theorem 7.24 Extreme value theorem ¶
    - depends_on lemma A.221 Counting identity ¶
    - depends_on lemma 6.19 Continuous argument along a path ¶
      - depends_on definition 6.20 Winding number ¶
      - depends_on lemma 6.22 Nearby loops wind alike ¶
    - depends_on theorem 7.40 Continuous functions are integrable ¶
      - depends_on corollary A.502 All iterated orders agree ¶
      - depends_on definition 16.11 Admissible class; functional ¶
      - depends_on definition 9.137 Elliptic integrals of the three kinds ¶
      - depends_on lemma A.501 Iterated integration over a box ¶
      - depends_on lemma 16.18 Fundamental lemma ¶
      - depends_on remark 7.128 What the derivations below take as given ¶
      - depends_on theorem 7.42 Fundamental theorem of calculus, I ¶
    - depends_on theorem 7.42 Fundamental theorem of calculus, I ¶ ↺
    - depends_on theorem 7.34 Rolle ¶
      - depends_on theorem 7.37 Cauchy mean value theorem and l'Hôpital's rule ¶
      - depends_on theorem 7.35 Mean value theorem ¶
      - depends_on theorem 7.38 Taylor's theorem with Lagrange remainder ¶
  - depends_on theorem 7.25 Heine–Cantor: uniform continuity ¶
    - depends_on lemma A.195 Riemann–Lebesgue, continuous compactly supported case ¶
      - depends_on lemma A.196 Inversion for a difference ¶
    - 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.499 What zero content buys ¶
      - depends_on corollary A.514 Degeneracy on a negligible set ¶ ↺
      - depends_on lemma A.500 Graphs and $C^{1}$ images have zero content ¶ ↺
      - depends_on theorem A.513 Change of variables ¶
    - 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.176 Helly–Bray ¶
      - depends_on corollary A.181 The converse ¶
    - depends_on lemma A.488 Small chords cut off small arcs ¶
      - depends_on theorem A.490 Equicontinuity of the normalised class ¶ ↺
    - depends_on lemma 14.34 The $n$-sphere is simply connected for $n \ge 2$ ¶
    - depends_on lemma 6.19 Continuous argument along a path ¶ ↺
    - depends_on remark 7.128 What the derivations below take as given ¶ ↺
    - depends_on theorem 7.40 Continuous functions are integrable ¶ ↺
    - depends_on theorem 7.109 Leibniz integral rule ¶
      - depends_on corollary 7.110 Variable limits of integration ¶
      - depends_on lemma 7.116 Functions vanishing on a regular zero set ¶
      - depends_on lemma A.607 A zero-free entire function is an exponential ¶
      - depends_on lemma A.599 One variable, complex coefficient ¶
      - depends_on proposition A.524 Rate of decay ¶
      - depends_on proposition A.521 Every derivative passes onto $g$ ¶
      - depends_on theorem 7.137 Helmholtz decomposition ¶
    - … 1 more
  - depends_on theorem A.173 Helly's selection theorem ¶
    - depends_on lemma A.176 Helly–Bray ¶ ↺
    - depends_on lemma A.175 Tightness makes the Helly limit proper ¶
depends_on corollary A.47 Cauchy completeness ¶
- depends_on theorem 7.8 Cauchy criterion ¶
  - 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 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 9.22 The exponential and its derivative ¶
      - depends_on theorem 9.8 Picard–Lindelöf ¶
    - depends_on proposition 7.48 Ratio test ¶
      - depends_on definition 7.53 $\ee$ ¶
      - depends_on definition 7.59 The real exponential ¶
      - depends_on definition 8.3 Exponential, sine, cosine ¶
      - depends_on lemma 7.72 Addition theorems ¶
      - depends_on proposition 7.85 Irrationality of $\pi$ ¶ ↺
      - depends_on theorem 7.51 Termwise differentiation ¶
    - 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 lemma 17.14 Uniformly bounded sine sums ¶
    - 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 ¶ ↺
depends_on corollary A.46 Monotone convergence ¶
- depends_on corollary A.47 Cauchy completeness ¶ ↺
- 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 remark 7.70 What the condition says ¶ ↺
    - depends_on definition 7.62 Real powers ¶
      - depends_on definition 7.90 The golden ratio ¶
      - depends_on proposition 7.63 Laws of real powers ¶
      - depends_on proposition 7.89 Irrationality of square roots ¶ ↺
    - 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 proposition 7.52 Alternating series test ¶
  - depends_on lemma 7.73 $\cos 2 < 0$ ¶
    - 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 proposition 7.49 Cauchy product ¶
  - depends_on lemma 7.60 Functional equation of the exponential ¶ ↺
  - 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 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 7.47 Comparison; absolute convergence ¶ ↺
- depends_on proposition 7.61 The logarithm ¶ ↺
- depends_on proposition 7.46 Geometric series ¶
  - depends_on lemma 7.86 Terminating or repeating decimals are rational ¶
    - depends_on proposition 7.91 The golden ratio ¶ ↺
    - depends_on proposition 7.85 Irrationality of $\pi$ ¶ ↺
  - depends_on lemma 7.54 Truncation error ¶
    - depends_on corollary 7.55 The decimal expansion ¶
    - depends_on proposition 7.56 Irrationality ¶
      - depends_on remark 7.94 Three numbers, two kinds ¶ ↺
    - depends_on remark 7.58 An identity, not a recipe ¶
      - depends_on remark 7.70 What the condition says ¶ ↺
  - depends_on lemma 7.139 A countable union of null sets is null ¶
    - depends_on proposition 7.143 The Hausdorff dimension never exceeds the box dimension ¶
    - depends_on proposition 7.141 The critical exponent, and the Hausdorff dimension ¶
      - depends_on proposition 7.143 The Hausdorff dimension never exceeds the box dimension ¶ ↺
    - depends_on remark 7.94 Three numbers, two kinds ¶ ↺
  - depends_on lemma A.288 The iteration converges ¶ ↺
  - depends_on proposition 7.48 Ratio test ¶ ↺
  - depends_on theorem 7.50 Power series; radius of convergence ¶ ↺
  - depends_on theorem 8.20 Taylor expansion ¶
    - depends_on lemma A.608 An entire function with a quadratic bound on its real part ¶
      - depends_on proposition A.609 The exponent is a quadratic polynomial ¶
    - depends_on proposition 8.23 Residue at a simple pole ¶
    - depends_on theorem A.598 Hudson ¶
    - depends_on theorem 8.21 Laurent expansion ¶
      - depends_on definition 8.22 Isolated singularities; residue ¶
      - depends_on theorem 8.24 Residue theorem ¶
- depends_on theorem 7.7 Bolzano–Weierstrass ¶ ↺

Neighborhood

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Edges

type	direction	node	provenance	where
`depends_on`	→	Cut	declared	appendices/A-long-proofs.tex:2963
`depends_on`	→	Order on $\R$	declared	appendices/A-long-proofs.tex:2963
`depends_on`	←	Archimedean property and density of $\Q$	declared	appendices/A-long-proofs.tex:3076
`depends_on`	←	Cauchy completeness	declared	appendices/A-long-proofs.tex:3109
`depends_on`	←	Monotone convergence	declared	appendices/A-long-proofs.tex:3095
`proves`	←	app:A-long-proofs@proof-29	declared	appendices/A-long-proofs.tex:2966