Sequences, series, summability Lemma
Every sequence has a monotone subsequence
Builds on SubsequenceMonotone sequence
Statement
Let be a real sequence. Then has a subsequence that is monotone.
Remarks
The usual proof uses peaks. An index is a peak if for all . If there are infinitely many peaks, the terms at the peaks form a decreasing subsequence. If there are finitely many peaks, start after the last one. Each index then has a later index with a larger term, and this gives a strictly increasing subsequence. With the monotone convergence theorem, the lemma gives the Bolzano–Weierstrass theorem.
Reading path
To understand this page from the ground up, read these 5 pages in order. The foundations come first.
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- Sequence definition
- Real sequence definition
- Subsequence definition
- Monotone sequence definition
- Every sequence has a monotone subsequence statement
Where this leads
165 pages build on this one.
- Power series are differentiable inside the radius of convergence
- A function with zero derivative is constant
- Natural logarithm
- A differentiable function is convex iff its derivative is increasing
- Taylor's theorem with the Cauchy remainder
- Euler–Mascheroni constant
- The logarithm grows slower than any power
- Derivative of sine
- 1 + x is at most exp x
- Reduction formula
- Rolle's theorem
- Weierstrass function
- First derivative test
- cos pi = -1
- sin x / x tends to 1
- Uniformly Cauchy sequence of functions
- A function with nonnegative derivative is increasing
- Inverse trigonometric functions
- Cauchy mean value theorem
- Leibniz formula for pi
- Integral of exp(-x) over the half line
- Pythagorean trigonometric identity
- log x is at most x - 1
- Inverse hyperbolic functions
- Integral of cosine
- Derivative of arctangent
- Harmonic numbers grow like log n
- First fundamental theorem of calculus
- Integral of the exponential function
- Power series of the exponential function
- Cauchy criterion for series
- Continuous functions are Riemann integrable
- Continuous image of a closed interval is a closed interval
- A function with positive derivative is strictly increasing
- Derivative of the logarithm
- Integration by parts
- Boundedness theorem
- Arcsine
- Mean value theorem
- Power series converge uniformly on smaller discs inside the radius of convergence
- pi
- Uniform convergence of derivatives allows term-by-term differentiation
- Integration by substitution
- The exponential function is convex
- Basel problem
- Trigonometric substitution
- The gamma function extends the factorial
- Second derivative test
- Derivative of arcsine
- Area under a curve
- Integral of sine
- Dirichlet integral
- Taylor's theorem with the Lagrange remainder
- Logarithm of a quotient
- Weierstrass approximation theorem
- L'Hôpital's rule
- Integral of 1/x
- Logarithm of a product
- Stirling's formula
- Bolzano–Weierstrass theorem
- The exponential function is strictly increasing
- Derivative of x^a
- Mean value theorem for integrals
- The integral of a continuous function is an antiderivative
- Fundamental theorem of calculus
- Weierstrass M-test
- Sine is 2π-periodic
- A function with bounded derivative is Lipschitz
- Integral of a power
- Darboux's theorem
- Irrationality of pi
- Continuity of the sum of a uniformly convergent series of functions
- pi is greater than 3
- Arccosine
- Logarithm is the inverse of the exponential
- A continuous nowhere differentiable function
- Power-reduction formula for cosine
- Derivative of tangent
- Normal convergence
- Power series of arctangent
- Term-by-term differentiation
- Real exponentiation
- Derivative of cosine
- Logarithm of a power
- Dirichlet's test
- Improper integral of x^p on an unbounded interval
- Continuous functions on a closed bounded interval are uniformly continuous
- Tangent half-angle substitution
- Integration by partial fractions
- x^2 sin(1/x) is differentiable with discontinuous derivative
- Taylor's theorem with the integral remainder
- Extreme value theorem
- Wallis product
- Dini's theorem
- Second fundamental theorem of calculus
- Absolutely convergent series converge
- Mercator series
- A function with nonnegative second derivative is convex
- Cauchy criterion for sequences
- The logarithm is strictly concave
- Arnold tongue
- Stone–Weierstrass theorem
- Lebesgue's theorem on differentiability of monotone functions
- Minimal curvature property of natural cubic splines
- Riemann zeta function as a series
- Stein's unbiased risk estimate
- L2 convergence of Fourier series
- Convergence of the Neumann series for Volterra kernels
- Weierstrass product for the gamma function
- Brent's method
- Stein's lemma
- Bertrand's test
- Mertens' theorems
- Peano kernel theorem
- Stein's example
- Bisection method
- Reduction of an initial value problem to a Volterra equation
- Ramanujan's series for 1/pi
- Difference quotient
- Continuous functional calculus
- Quadratic convergence of Newton's method
- Failure of Aizerman's conjecture
- Frullani integral
- Darboux function
- Gauss's test
- Log-convexity of the gamma function
- Mean value theorem for divided differences
- Blancmange curve
- Uniform convergence of absolutely summable Fourier series
- Differentiation operator as inverse of integration
- Completeness of the real numbers
- Logarithmically convex function
- Cauchy formula for repeated integration
- Lidskii's trace formula for integral operators
- Approximation of a kernel by degenerate kernels
- Reduction of a first-kind Volterra equation to the second kind
- Liouville–Neumann series
- Interpolation error formula
- Density of trigonometric polynomials in continuous functions
- Borel's lemma
- Viète's formula
- Nested intervals theorem
- Compactness of integral operators
- Cantor function
- Commutative Gelfand–Naimark theorem
- Fourier basis of L2 of the circle
- Jentzsch's theorem on positive kernels
- Mercer's theorem
- Error bound for the trapezoidal rule
- Bohr–Mollerup theorem
- Volterra's function
- Noncommutative geometry
- An infinite product converges iff its series of logarithms converges
- Quintic equation
- Maximum and minimum
- Descartes' rule of signs
- Trigonometric integrals
- Basel problem via Fourier series
- Abel's summation formula
- Unique solvability of Volterra equations of the second kind
- Abel–Dini theorem
- Decay of Fourier coefficients and smoothness
- Barnes G-function
- Chudnovsky algorithm
- Runge's phenomenon
Further reading
About this page
| Tag | 0MHGJRZMTM |
|---|---|
| URI | m2://0MHGJRZMTM |
| Kind | statement, lemma |
| Writing | W0 stub |
| MSC | 40A05 |
| Status | This copy of the site has no build data. Run m2 status 0MHGJRZMTM. |
| Uses | Subsequence, Monotone sequence |
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