definition A.36 Cut

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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 definition A.41 Addition ¶
  • depends_on definition A.43 Multiplication ¶
    • depends_on lemma A.44 $\R$ is an ordered field ¶
  • depends_on lemma A.44 $\R$ is an ordered field ¶ ↺
  • depends_on lemma A.42 $(\R, +)$ is an ordered abelian group ¶
    • depends_on lemma A.44 $\R$ is an ordered field ¶ ↺
• depends_on definition A.43 Multiplication ¶ ↺
• depends_on definition A.38 Order on $\R$ ¶
  • depends_on lemma A.42 $(\R, +)$ is an ordered abelian group ¶ ↺
  • depends_on lemma A.39 The order is total ¶
  • depends_on theorem A.40 Least-upper-bound property ¶
    • depends_on corollary A.45 Archimedean property and density of $\Q$ ¶ ↺
    • depends_on corollary A.47 Cauchy completeness ¶
      • depends_on theorem 7.8 Cauchy criterion ¶
        • depends_on lemma A.288 The iteration converges ¶
        • depends_on lemma A.454 Arzelà–Ascoli on an interval ¶ ↺
        • depends_on proposition 7.47 Comparison; absolute convergence ¶
        • 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 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 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 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 7.72 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 lemma 7.54 Truncation error ¶
        • depends_on lemma 7.139 A countable union of null sets is null ¶
        • 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 theorem 7.7 Bolzano–Weierstrass ¶ ↺
• depends_on lemma A.37 Elementary properties of cuts ¶
  • depends_on lemma A.42 $(\R, +)$ is an ordered abelian group ¶ ↺
  • depends_on lemma A.39 The order is total ¶ ↺
• depends_on theorem A.40 Least-upper-bound property ¶ ↺