Field theory and polynomials Proof
Proof of the Archimedean property in an Archimedean ordered field
Builds on Archimedean ordered fieldOrdered field
Proof
- Let be an Archimedean ordered field and let .
- First, in . The order is total, so or .
- If , add to both sides, using the first axiom of an ordered field. This gives .
- Then the second axiom gives . So in both cases.
- In a field . Hence .
- Apply the definition of an Archimedean field with . It gives a natural number with .
- Here , with terms, is the element of . So .
Idea
The definition says that copies of any positive add up past any . Taking , which is positive in each ordered field, gives the claim.
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- Proof of the inverse function theorem by the contraction mapping principle
- Incompressible flow
- Kharitonov's theorem
- Proof of the polar decomposition from the singular value decomposition
- Poincaré–Bendixson theorem for a bounded region without equilibria
- Homoclinic bifurcation
- Maxwell relations
- Duality of Lp spaces
- Random vibration
- Born rule
- Compactness of integral operators
- Dynamical friction
- Borel functional calculus
- Galerkin method
- Wiener–Hopf method
- Mixing length theory
- Kepler's equation
- Compactness of families of harmonic functions
- Proof that 1 + x is at most exp x
- Proof of the formula for the characteristic function of the normal distribution
- Bernstein inequalities
- Interchange of summation order
- Torsion of a curve
- Minkowski inequality
- Proof that a function with positive derivative is strictly increasing
- Proof of integration by substitution
- Gradient estimates for harmonic functions
- Proof of the second Borel–Cantelli lemma
- Kepler orbit
- Frank-Kamenetskii theory of thermal explosion
- Proof that sin x / x tends to 1 as a derivative at 0
- Osgood's lemma
- Earth's magnetic field
- Oseledets theorem
- Neumann formula for mutual inductance
- Cantor function
- Veblen–Young theorem
- Electric displacement field
- Proof of Ptolemy's theorem by complex numbers
- Configuration space
- Commutative Gelfand–Naimark theorem
- Carathéodory metric
- Schmidt decomposition
- Fourier basis of L2 of the circle
- Brody hyperbolicity
- Hyperbolic set
- Parametric oscillator
- Symplectic integrator
- Proof of the second fundamental theorem of calculus
- Schwarz reflection principle
- Harnack's principle
- Operator monotone function
- Proof of global existence for globally Lipschitz vector fields by steps of uniform length
- Hill sphere
- Proof of the spectral theorem for normal matrices by deflation
- Proof of the formula for the Laplace transform of a derivative by integration by parts
- Jentzsch's theorem on positive kernels
- Proof of Taylor's theorem with the Lagrange remainder by the Cauchy mean value theorem
- Proof of the Poisson limit theorem
- Poisson integral formula
- Kneser's theorem on solution sets
- Proof of Liouville's theorem on Diophantine approximation
- Nörlund–Rice integral
- Gauss's spherical harmonic analysis of the geomagnetic field
- Mercer's theorem
- Spherical astronomy
- Proof of Heron's formula from the law of cosines
- Littlewood subordination theorem
- Proof of cosh^2 - sinh^2 = 1
- Proof of the spectral theorem for Hermitian matrices
- Lagrange multipliers for constraints
- Roche limit
- Proof of triangularisation by induction on the size
- Noisy-channel coding theorem
- Proof that angles in the same segment are equal
- Bifurcation theory
- Proof of term-by-term integration
- Castigliano's method
- Error bound for the trapezoidal rule
- Duhamel's principle
- Gelfand–Bratu problem
- Proof of Taylor's theorem in several variables by restriction to a line
- Proof of the power series of arctangent by integrating a finite geometric sum
- Lagrange multipliers on Banach spaces
- Anharmonic oscillator
- Proof of the volume of a ball in R^3 from the n-dimensional formula
- Somigliana identity
- Kempner series
- Generalized Lotka–Volterra equation
- Magnetic helicity
- Proof of the impossibility of trisecting the angle
- Proof that harmonic numbers grow like log n
- Annihilator method
- Mechanical similarity
- Green's identities
- Finite difference coefficients
- Least harmonic majorant
- Proof that sine and cosine are 2π-periodic with smallest period 2π
- Bohr–Mollerup theorem
- Schönberg–Chandrasekhar limit
- Magnetotellurics
- First main theorem of Nevanlinna theory
- Proof of the isolated zeros theorem by power series
- Liouville's theorem for harmonic functions
- Quantum computing
- Proof of the derivative of sine by term-by-term differentiation
- Wave function collapse
- Proof of the additivity of line integrals under concatenation
- Flory–Huggins solution theory
- Convolution theorem
- Proof of Taylor's theorem with the Peano remainder
- Proof that the exponential of a sum of commuting matrices is the product of the exponentials
- Geopotential
- Elastic pendulum
- Proof of the solutions for repeated roots
- Principal component analysis
- Proof of the derivative of x^a by the chain rule
- Liouville's theorems on elliptic functions
- Proof of the fundamental theorem of contour integration
- Proof of Goursat's theorem by repeated subdivision
- N-body simulation
- Proof of Grönwall's inequality
- Proof that uniform limits of holomorphic functions are holomorphic, by Morera's theorem
- Proof of the solutions for distinct real roots
- Proof of angle-angle similarity with the law of sines
- Volterra's function
- Celestial mechanics
- Proof of L'Hôpital's rule from the Cauchy mean value theorem
- Iterated exponentiation
- Vogt–Russell theorem
- Hu–Washizu principle
- Mild solution
- Eddington luminosity
- Whitney extension theorem
- Heun's method
- Proof of the Poincaré lemma for one-forms on convex sets
- Hubbard–Stratonovich transformation
- Proof of the existence and uniqueness of the principal value of the argument
- Proof of the mean of the Poisson distribution
- Initial value formulation of general relativity
- Proof of continuous dependence on initial data by Grönwall's inequality
- Harmonic functions on Riemann surfaces
- Gauss's law
- Noncommutative geometry
- Hill sphere
- Proof of the polar form of a complex number
- Phase space
- Proof of the identity theorem by connectedness
- Proof of the integral of sine
- An infinite product converges iff its series of logarithms converges
- Elastic collision
- Proof of the maximum modulus principle by Cauchy's integral formula
- Quintic equation
- Axiom A
- Proof that arguments add under multiplication
- Inertia tensor and principal axes
- Proof of the exponential of a sum by the Cauchy product
- Barnes–Hut simulation
- Bidiagonalization
- Vis-viva equation
- Covariant formulation of classical electromagnetism
- Proof of the existence and uniqueness of Laurent expansions on an annulus
- Ladyzhenskaya–Babuška–Brezzi condition
- Maximum and minimum
- Proof of Stirling's formula by a monotone sequence and the Wallis product
- Duffing equation
- Classical mechanics
- Proof of the Casorati–Weierstrass theorem by Riemann's theorem
- Kapitza's pendulum
- Proof of the Pythagorean trigonometric identity by differentiation
- Work–energy theorem
- Stellar dynamics
- Milloux problem
- Kelly criterion
- Proof that the exponential function is its own derivative, by term-by-term differentiation
- Liouville's theorem for planar harmonic functions
- Liénard–Chipart criterion
- Conservation of momentum
- Gauss's law for potentials
- Proof of the second derivative test in several variables
- Proof that holomorphic functions are analytic
- Proof that the divergence of a curl is zero
- Proof of the power series of the exponential function
- Variable-mass system
- Maximum principle for harmonic functions
- Parker instability
- Descartes' rule of signs
- Trigonometric integrals
- Surface plasmon polariton
- Cauchy momentum equation
- Tautological one-form
- Indirect utility function
- Magnus expansion
- Extremal length
- Geomagnetic secular variation
- Bertrand's theorem
- Central force
- Einstein radius
- Integral transforms of tempered distributions
- Schur product theorem
- Hadamard three-circle theorem for subharmonic functions
- Fundamental theorem of space curves
- Lindelöf's maximum principle
- Helmholtz decomposition of electromagnetic fields
- Proof that the matrix exponential solves a linear system
- Basel problem via Fourier series
- Hadamard three-circle theorem
- Schwarz–Pick theorem
- Singular values of a compact operator
- Sitnikov problem
- Proof of the Gauss–Wantzel theorem
- Abel's summation formula
- Exponential stability
- Proof of the Mercator series by term-by-term differentiation and Abel's theorem
- Spherical pendulum
- Proof of the symmetry of second derivatives by a second difference
- Jury stability criterion
- Laplace transform of an integral
- Graph embedding
- Fixation probability
- Observable
- Proof of the formula for the moment-generating function of the normal distribution
- Pesin theory
- Reptation
- Slice theorem
- Proof that a function with zero derivative on an open interval is constant
- Duhamel's principle for heat conduction
- Schmidt's singular value decomposition of a kernel
- Proof that the determinant of a matrix exponential is the exponential of the trace
- Proof that the gamma function extends the factorial
- Impossibility of the classical construction problems
- Free-fall time
- Besicovitch set of measure zero
- Proof of the first derivative test
- Jacobi ellipsoid
- Proof of the boundedness theorem by Bolzano–Weierstrass
- Lagrange points
- Harmonic measure as exit distribution
- Van der Pol oscillator
- Two-body relaxation time
- Stable manifold theorem
- Proof that the exterior derivative squares to zero
- Unique solvability of Volterra equations of the second kind
- Midpoint method
- Hadamard three-lines theorem
- Parker spiral
- Proof of the Picard–Lindelöf theorem in integral form
- Picard's theorem for systems with continuous partial derivatives
- Fubini's theorem for series
- Proof of the p-series test by Cauchy condensation
- Parabolic and hyperbolic Riemann surfaces
- Poisson integral for the half-space
- Retarded potential
- Polar decomposition of the deformation gradient
- Proof of the weak law of large numbers by characteristic functions
- Anosov diffeomorphism
- Analytic capacity
- Brewster's angle
- Laplace transform of the exponential function
- Differentiable curve
- Proof that C^k functions are characterised by continuous derivatives
- Proof of the Chapman–Kolmogorov equation
- Spectral theorem for bounded normal operators
- Poynting's theorem
- Newtonian limit
- Adjoint state method
- Proof that continuous partial derivatives imply differentiability
- Strong gravitational lensing
- Lindelöf's theorem
- Harmonic space
- Proof of the vector calculus identities
- Hadamard's two-constants theorem
- Stability conjecture
- Prime end
- Monte Carlo methods in finance
- State-transition matrix
- Proof of the fundamental theorem of calculus
- Hidden-variable theory
- Channel capacity of a quantum channel
- Proof that the real part of a holomorphic function is harmonic
- Bôcher's theorem
- Einstein ring
- Madelung equations
- Circular restricted three-body problem
- Gain scheduling
- Jefimenko's equations
- Function of exponential order
- Faraday's law of induction
- Proof of the mean value theorem in several variables
- Proof of the Picard–Lindelöf theorem
- Proof of the mean value theorem from Rolle's theorem
- Evolute
- Barnes G-function
- Proof of the addition formulas for hyperbolic functions
- Groundwater flow equation
- Hill differential equation
- Proof of the logarithm of a product
- Automorphism group of a bounded domain is a Lie group
- Proof of the Wallis product by integrals of powers of sine
- International Geomagnetic Reference Field
- Circle average
- Rellich–Kondrachov theorem
- Proof of the integral of 1/x
- Proof that the curl of a gradient is zero
- Proof that pi is greater than 3 by the alternating series estimate
- Proof of the classification of isometries of the plane
- Worm-like chain
- Niven's proof of the irrationality of pi
- Magnetic vector potential
- Morrey's inequality
- Osculating circle
- Chudnovsky algorithm
- Bullen parameter
- Proof of Liouville's theorem by Cauchy's estimates
- Four-vertex theorem
- Euler integral for the gamma function
- Diversity index
- Normal form
- Roche lobe
- Hartogs lemma for subharmonic functions
- Proof of the inscribed angle theorem by complex numbers
- Proof that continuous functions are Riemann integrable
- Proof of the mean value inequality
- Frenet–Serret formulas
- Runge's phenomenon
- Proof that the exponential function is strictly convex
- Proof of Hurwitz's theorem by Rouché's theorem
- Whitney–Graustein theorem
- Inductance
- Proof of the exterior angle theorem from the angle sum
- Montel's theorem
- Euler–Lagrange equation
- Residue calculus for Laplace inversion
Further reading
- Web https://en.wikipedia.org/wiki/Archimedean_property, Ordered fields
About this page
| Tag | RCMSYW1KBX |
|---|---|
| URI | m2://RCMSYW1KBX |
| Kind | proof |
| Writing | W0 stub |
| MSC | 12J15 |
| Proves | Archimedean property |
| Uses | Archimedean ordered field, Ordered field |
| Changed | 4 November 2026 |
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