Sequences, series, summability Sequences and limits Theorem
Bolzano–Weierstrass theorem
Builds on Bounded sequenceSubsequenceConvergent sequenceEvery sequence has a monotone subsequenceMonotone convergence theorem for sequences
Statement
Let be a bounded sequence of real numbers. Then has a subsequence that converges to a real number.
Remarks
The proof takes a monotone subsequence. It is bounded, so it converges by the monotone convergence theorem. The limit of the subsequence lies in each closed interval that contains all terms, by the preservation of non-strict inequalities. The theorem holds in for each . It says that each closed and bounded subset of is sequentially compact. The theorem fails in . The decimal approximations of are bounded, and no subsequence has a limit in .
History
Bernard Bolzano proved the result in 1817, as a lemma for the intermediate value theorem. Karl Weierstrass proved it again about fifty years later and made it a central tool of analysis.
Reading path
To understand this page from the ground up, read these 10 pages in order. The foundations come first.
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- Sequence definition
- Real sequence definition
- Subsequence definition
- Bounded sequence definition
- Limit of a sequence definition
- Monotone sequence definition
- Convergent sequence definition
- Monotone convergence theorem for sequences statement
- Every sequence has a monotone subsequence statement
- Bolzano–Weierstrass theorem statement
Where this leads
164 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
- 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 | 0KDJD249DT |
|---|---|
| URI | m2://0KDJD249DT |
| Kind | statement, theorem |
| Writing | W1 stated |
| MSC | 40A05 |
| Status | This copy of the site has no build data. Run m2 status 0KDJD249DT. |
| Uses | Bounded sequence, Subsequence, Convergent sequence, Every sequence has a monotone subsequence, Monotone convergence theorem for sequences |
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