Bounded sequence

Bounded sequence
Tag	`05FQ3VGAR9`
URI	`m2://05FQ3VGAR9`
Kind	definition
Writing	W0 stub
MSC	40A05
Defines	Bounded sequence
Uses	Real sequence

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Definition

Let $(a_n)$ be a real sequence. The sequence is bounded above if there is a real number $M$ with $a_n \le M$ for all $n \in \mathbb{N}$ . The sequence is bounded below if there is a real number $m$ with $a_n \ge m$ for all $n \in \mathbb{N}$ . The sequence is bounded if it is bounded above and bounded below. Equivalently, $(a_n)$ is bounded if there is a real number $K \ge 0$ with $|a_n| \le K$ for all $n \in \mathbb{N}$ .

Remarks

The sequence $a_n = (-1)^n$ is bounded, with $K = 1$ . The sequence $a_n = n$ is bounded below by $0$ , but it is not bounded above. A sequence is a function on $\mathbb{N}$ , so a bounded sequence is a bounded function on $\mathbb{N}$ . Each convergent sequence is bounded, as the node convergent sequences are bounded states.

Explanations

L1 human

A sequence is bounded when all its terms stay inside one fixed interval $[-K, K]$ . The terms can move up and down, but they never escape from that interval.

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Sequence definition
Real sequence definition
Bounded sequence definition

What depends on this

172 nodes use this node, directly or through other nodes.

Power series are differentiable inside the radius of convergence 002X7EK91H
A function with zero derivative is constant 003QDZ4T16
Natural logarithm 0044CPHJBV
A differentiable function is convex iff its derivative is increasing 005JT76HCK
Taylor's theorem with the Cauchy remainder 005Q84BMAD
Euler–Mascheroni constant 00C5P9JZBK
The logarithm grows slower than any power 00DGYY10FV
Derivative of sine 00R4MWW532
1 + x is at most exp x 012C7GFF20
Reduction formula 012RYET5E2
Riemann's criterion for integrability 017DH8WXTE
Rolle's theorem 0194DY35BG
Integral test 019QZ6RRVH
Weierstrass function 01A0MK1730
First derivative test 01A2EGXJEM
cos pi = -1 01A8JGR4JM
sin x / x tends to 1 01QSSKS5C1
Uniformly Cauchy sequence of functions 01XAY17QA3
Darboux and Riemann integrals agree 021XJAQVDC
A function with nonnegative derivative is increasing 026BJBQNGJ
Angle addition formulas 028XEQCE88
Inverse trigonometric functions 02HTDK1FBD
Power series of sine 02KTRHQMHN
Cauchy mean value theorem 02SYZAWM2K
Decreasing sequence bounded below converges 02V2GYAC8P
Leibniz formula for pi 037FD5NBRT
Integral of exp(-x) over the half line 039JAF7ZAN
Pythagorean trigonometric identity 03H7WZ4T3K
Limit of a quotient of sequences 03S6GWWKEZ
The Dirichlet function is not Riemann integrable 03Z4JC4YRJ
log x is at most x - 1 04CBRA4JRK
Inverse hyperbolic functions 04D7CHT5V1
Integral of cosine 04WTBQ7APQ
Monotone functions are Riemann integrable 04YKAYDQPW
Comparison test 04Z5G0THN6
Root test 05H4DDDYBZ
Derivative of arctangent 05P4B4Q22V
Harmonic numbers grow like log n 05S2GSPMGA
First fundamental theorem of calculus 063H5KJC69
Arctangent 065XW2V6C3
Radius of convergence 0660QX1VPR
Integral of the exponential function 06D0EWR2DV
Power series of the exponential function 06JZGMYTYD
Cauchy criterion for series 06PVXMY0TH
cosh^2 - sinh^2 = 1 06QCTRFBM0
Continuous functions are Riemann integrable 06SESV7GQY
Tangent function 06T0HS14PP
Continuous image of a closed interval is a closed interval 06ZARTQTQE
Limit of a product of sequences 0734RSVF2V
Continuity of the inverse of a strictly monotone continuous function 07BY47HG5S
Cauchy–Hadamard theorem 07E5MB1CA2
A function with positive derivative is strictly increasing 07J5X2T1ZX
Derivatives of the hyperbolic functions 07T2TKGZ4J
Derivative of the logarithm 07X1QA6BRW
Divergence part of the ratio test 08AM0KRCZ1
Power series of cosine 08W5RN375X
Integration by parts 08Y1BBQTK9
Upper and lower Darboux integrals 09FQT057T7
Boundedness theorem 0ATVXC3NCC
Arcsine 0B89D93YA1
Cauchy condensation test 0B96KMDDD1
Mean value theorem 0BARQY52N5
Uniform limits commute with integration 0BWGW68B66
Power series converge uniformly on smaller discs inside the radius of convergence 0BWXV7GADW
pi 0C1VYVNV2N
Polynomial functions are continuous 0C5TP6RYSV
Abel's test 0CN97GADE5
Uniform convergence of derivatives allows term-by-term differentiation 0CS42KEM5C
Exponential growth beats polynomial growth 0CT6CBGTYY
Integration by substitution 0D37QFPEZE
The exponential function is convex 0D7GRZ2ADX
Gaussian integral 0DFT1GP1JP
Basel problem 0DMY75D6FV
Exponential of a sum 0E8HNVBCSP
Trigonometric substitution 0EAR3KNRTN
Hyperbolic cosine 0EQFWKJ37J
The exponential function is its own derivative 0EX9GFJ0B3
The gamma function extends the factorial 0FDCW77H44
e as the limit of (1 + 1/n)^n 0FEMF8JKAF
Monotone convergence theorem for sequences 0FM6N4R0VB
Comparison test for improper integrals 0FR1BDKK10
Second derivative test 0FZFHK97P7
Sum of the reciprocals of the factorials 0G4D3ZKTP8
Cauchy sequences are bounded 0G6D8E7CTA
Derivative of arcsine 0GA6G43EA9
Addition formulas for hyperbolic functions 0GJZCE87WP
Quotient of continuous functions is continuous 0GR2APWTAC
Decimal expansion as a series 0GT85FBPTS
Hyperbolic sine 0GWD21EWRW
Area under a curve 0GXA8Z205E
Integral of sine 0GXCXCX3QN
Dirichlet integral 0GZNSHMG4Q
Limit inferior 0H4R3RESGV
Taylor's theorem with the Lagrange remainder 0H5ZAZCX2T
Binomial series 0HB0XKDXHH
A sequence converges iff its liminf and limsup agree 0HM732ZNVC
Limit comparison test 0HW7BNHHGF
Existence of nth roots by the intermediate value theorem 0J29GM30Z7
Hyperbolic functions 0J6M03WSS6
Logarithm of a quotient 0J7PHRBNNS
Ratio formula for the radius of convergence 0JBMYAA9QH
Alternating harmonic series 0JQ5Y3WWVE
Weierstrass approximation theorem 0JQE130DHW
L'Hôpital's rule 0JWT4PPQ7B
Integral of 1/x 0K1N1Y4EGZ
Logarithm of a product 0K2FNY6DYV
Stirling's formula 0KC0F51T6A
Bolzano–Weierstrass theorem 0KDJD249DT
One-sided limits of a monotone function exist 0KJXHN0TW0
The exponential function is strictly increasing 0KN2V7840E
Radius of convergence of the binomial series 0KX1MXJ5MR
Derivative of x^a 0KY08TWDMQ
Derivative of an inverse function 0M0RZ3HSB0
e 0MVCFMW7S4
Hyperbolic tangent 0MWYQTZ07Z
A series of nonnegative terms converges iff its partial sums are bounded 0MX1HZBBXJ
Mean value theorem for integrals 0N91N98EG5
The integral of a continuous function is an antiderivative 0NC2A3HWGT
Fundamental theorem of calculus 0NHGY0N0JB
Weierstrass M-test 0NX5NWAFC9
Sine is 2π-periodic 0P4NQD746K
A function with bounded derivative is Lipschitz 0P64VE79X3
Limit superior 0P8BA7T7S5
The exponential function is positive 0P9BNBW2W3
Alternating series test 0PK8QH1W1V
The exponential series has infinite radius of convergence 0PVJRACB9H
Integral of a power 0Q3HYFT9RQ
Darboux's theorem 0Q5A47J48D
Irrationality of pi 0QHZX1PK8V
Ratio test 0QNY6ZBXPN
Upper and lower Darboux sums 0QRQVH0CRW
Continuity of the sum of a uniformly convergent series of functions 0R0MYT0BJD
pi is greater than 3 0R7SVNYTJS
Arccosine 0RDP2HEFEW
Logarithm is the inverse of the exponential 0RKW2X63YR
Exponential function 0RMVS7RXZX
A continuous nowhere differentiable function 0RXK2Z808Q
Power-reduction formula for cosine 0S049HG1W5
Convergent sequences are bounded 0S4JF619V7
Indeterminate form 0S7G92KKA7
Derivative of tangent 0SAMR34WXC
Normal convergence 0T0EJDGHMB
Power series of arctangent 0T32ME0A6H
Term-by-term differentiation 0TJ8QBJX5K
Real exponentiation 0TV07AQT62
Derivative of cosine 0V2DV29EGT
Gamma function 0V3FZMVB4K
Logarithm of a power 0V948DE5KA
Dirichlet's test 0V9ME8GQE7
Improper integral of x^p on an unbounded interval 0VGKB2NC4T
Product of continuous functions is continuous 0VGMHCF4SC
Continuous functions on a closed bounded interval are uniformly continuous 0W0TZDRF9C
Irrationality of e 0W4GQSG7B6
Tangent half-angle substitution 0W8ETY2AZ6
Sine and cosine 0WB57D9488
Integration by partial fractions 0WBTWFXJQG
Rational function 0WCZH4AFMT
x^2 sin(1/x) is differentiable with discontinuous derivative 0WDAKKN4NC
Taylor's theorem with the integral remainder 0WJGR81HBD
Extreme value theorem 0WS1A8E2P9
A monotone function has countably many discontinuities 0WYTM4C9QG
Wallis product 0XCYA4YERS
Dini's theorem 0XRRYF1HPE
Second fundamental theorem of calculus 0XYAFCN0PT
Absolutely convergent series converge 0Y123X48GA
Term-by-term integration 0YBY3VYX0K
Mercator series 0YZXNE3HFR
A function with nonnegative second derivative is convex 0Z41C17TGP
Interval of convergence 0Z5R9TVWQH
Cauchy criterion for sequences 0ZA4AX4MW3
p-series test 0ZNSSK8ZK1
The logarithm is strictly concave 0ZS54B9GJF

Sources

Web https://en.wikipedia.org/wiki/Sequence, Bounded
Web https://en.wikipedia.org/wiki/Bounded_function, lead section

What links here

Used by

Radius of convergence 0660QX1VPR
Abel's test 0CN97GADE5
Monotone convergence theorem for sequences 0FM6N4R0VB
Cauchy sequences are bounded 0G6D8E7CTA
Limit inferior 0H4R3RESGV
Bolzano–Weierstrass theorem 0KDJD249DT
Limit superior 0P8BA7T7S5
Upper and lower Darboux sums 0QRQVH0CRW
Convergent sequences are bounded 0S4JF619V7