definition A.38 Order on $\R$

open in the book · appendices/A-long-proofs.tex:2933 · p. 2810

ground object -- no derivation owed

Rests on

depends_on definition A.36 Cut ¶

Supports

depends_on lemma A.42 $(\R, +)$ is an ordered abelian group ¶
- depends_on lemma A.44 $\R$ is an ordered field ¶
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 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 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 7.89 Irrationality of square roots ¶
      - depends_on proposition 7.91 The golden ratio ¶
  - 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.455 Compact embedding of $W^{1,r}(a,b)$ into the continuous functions ¶
    - 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 theorem 7.40 Continuous functions are integrable ¶
      - depends_on theorem 7.42 Fundamental theorem of calculus, I ¶
      - depends_on theorem 7.34 Rolle ¶
    - depends_on theorem 7.25 Heine–Cantor: uniform continuity ¶
      - depends_on lemma A.195 Riemann–Lebesgue, continuous compactly supported case ¶
      - depends_on lemma A.500 Graphs and $C^{1}$ images have zero content ¶
      - depends_on lemma A.499 What zero content buys ¶
      - depends_on lemma A.512 The boundary strip is thin ¶
      - depends_on lemma A.73 Differentiation under the integral sign ¶
      - depends_on lemma A.176 Helly–Bray ¶
      - depends_on lemma A.488 Small chords cut off small arcs ¶
      - 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 ¶
      - … 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.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 7.48 Ratio test ¶
      - depends_on proposition 8.4 Euler's formula ¶
      - depends_on theorem 7.50 Power series; radius of convergence ¶
      - 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 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 proposition 7.52 Alternating series test ¶
    - depends_on lemma 7.73 $\cos 2 < 0$ ¶
      - depends_on definition 7.74 $\pi$ ¶
  - 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 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 7.72 Addition theorems ¶
      - depends_on proposition 7.79 The polygon recursion ¶ ↺
      - depends_on proposition 7.77 Special values, periodicity, and the kernel ¶ ↺
      - 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 7.47 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 7.58 An identity, not a recipe ¶
    - 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 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 8.23 Residue at a simple pole ¶
      - depends_on theorem A.598 Hudson ¶
      - depends_on theorem 8.21 Laurent expansion ¶
  - depends_on theorem 7.7 Bolzano–Weierstrass ¶ ↺

Neighborhood

Every logical edge within two steps of this node.

declared and complete
partly declared
a check failed
not graded
declared in the source
inferred from structure

Edges

type	direction	node	provenance	where
`depends_on`	→	Cut	declared	appendices/A-long-proofs.tex:2937
`depends_on`	←	$(\R, +)$ is an ordered abelian group	declared	appendices/A-long-proofs.tex:3005
`depends_on`	←	The order is total	declared	appendices/A-long-proofs.tex:2943
`depends_on`	←	Least-upper-bound property	declared	appendices/A-long-proofs.tex:2963