Limit of a sequence

Limit of a sequence
Tag	`0XKWSSY8P2`
URI	`m2://0XKWSSY8P2`
Kind	definition
Writing	W1 stated
MSC	40A05
Defines	Limit of a sequence
Uses	Real sequence

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Definition

Let $(a_n)$ be a real sequence and let $L$ be a real number. The sequence $(a_n)$ converges to $L$ if the following holds for each open interval $U$ that contains $L$ . There is a natural number $N$ such that $a_n \in U$ for all $n \ge N$ . Then $L$ is a limit of $(a_n)$ , with the notation $\lim_{n \to \infty} a_n = L$ or $a_n \to L$ as $n \to \infty$ .

Remarks

The epsilon-N definition of the limit of a sequence is an equivalent form, with intervals $(L - \varepsilon, L + \varepsilon)$ . A sequence has at most one limit, by the uniqueness of limits. So the notation $\lim_{n \to \infty} a_n$ is well defined. The same definition works in each topological space, with open neighbourhoods of $L$ in place of open intervals. In Mathlib, the limit is the statement Filter.Tendsto a Filter.atTop (nhds L).

Explanations

L1 human

The sequence $(a_n)$ tends to $L$ when its terms come close to $L$ and stay close. However small a window around $L$ you choose, from some index on, all terms are inside the window. For example, $1/(n+1)$ tends to $0$ .

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Sequence definition
Real sequence definition
Limit of a sequence definition

What depends on this

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

Sequential characterization of limits 000MXHEY22
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
Rolle's theorem 0194DY35BG
Integral test 019QZ6RRVH
Weierstrass function 01A0MK1730
First derivative test 01A2EGXJEM
cos pi = -1 01A8JGR4JM
Divergent sequence 01HR2AH0BX
sin x / x tends to 1 01QSSKS5C1
Geometric series 01X5CVV1Q5
Uniformly Cauchy sequence of functions 01XAY17QA3
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
Conditional convergence 02YRQXZDCS
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
Comparison test 04Z5G0THN6
Pointwise convergence 056FSKG32J
Root test 05H4DDDYBZ
Derivative of arctangent 05P4B4Q22V
Harmonic numbers grow like log n 05S2GSPMGA
First fundamental theorem of calculus 063H5KJC69
Arctangent 065XW2V6C3
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
Algebra of limits 07XBM3A9J5
Divergence part of the ratio test 08AM0KRCZ1
Uniform convergence 08FPABDQ4C
Uniform convergence on a set 08HJRTG5N7
The sequence 1/n tends to zero 08QAX4H1PF
Power series of cosine 08W5RN375X
Integration by parts 08Y1BBQTK9
Epsilon-N definition of the limit of a sequence 09G6KHS6GY
sin(1/x) has no limit at 0 0AFWQZ1PKF
Cauchy product of absolutely convergent series 0AN28XQ45Z
Boundedness theorem 0ATVXC3NCC
The nth root of n tends to 1 0AVEYZ2T0Q
Convergent sequence 0AZ3R52593
Arcsine 0B89D93YA1
Cauchy condensation test 0B96KMDDD1
Mean value theorem 0BARQY52N5
Mertens' theorem on Cauchy products 0BJSWZJG18
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
Convergent series 0CVVKY7GY2
Harmonic series 0CXTMAY5PX
Integration by substitution 0D37QFPEZE
The exponential function is convex 0D7GRZ2ADX
Gaussian integral 0DFT1GP1JP
Basel problem 0DMY75D6FV
Riemann rearrangement theorem 0E7VBYGFF7
Exponential of a sum 0E8HNVBCSP
Harmonic series diverges 0E9M24KS9J
Trigonometric substitution 0EAR3KNRTN
Hyperbolic cosine 0EQFWKJ37J
The exponential function is its own derivative 0EX9GFJ0B3
x sin(1/x) is continuous at 0 0EZMAY1FQD
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
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
Taylor's theorem with the Lagrange remainder 0H5ZAZCX2T
Binomial series 0HB0XKDXHH
Thomae's function 0HJ28H9Y95
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
Rearrangements of an absolutely convergent series have the same sum 0J76TP9FGD
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
Sum of a series 0K0H5E3Z5N
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
Linearity of differentiation 0M0TKFRX2A
Derivative of a polynomial 0MH1WK89PP
Series of functions 0MRE6XRE5V
e 0MVCFMW7S4
Sequential continuity 0MVDQEAB60
Exponentials dominate powers in sequences 0MVTCFW42K
Hyperbolic tangent 0MWYQTZ07Z
A series of nonnegative terms converges iff its partial sums are bounded 0MX1HZBBXJ
Limits preserve non-strict inequalities 0N32V0E3XJ
Mean value theorem for integrals 0N91N98EG5
Divergent series 0NBRP6093V
The integral of a continuous function is an antiderivative 0NC2A3HWGT
Fundamental theorem of calculus 0NHGY0N0JB
Triangle inequality for integrals 0NR82KMYKD
Weierstrass M-test 0NX5NWAFC9
Sine is 2π-periodic 0P4NQD746K
A function with bounded derivative is Lipschitz 0P64VE79X3
The exponential function is positive 0P9BNBW2W3
Continuity is equivalent to sequential continuity 0PAQM3H57X
Alternating series test 0PK8QH1W1V
The exponential series has infinite radius of convergence 0PVJRACB9H
Stolz–Cesàro theorem 0Q3AKZV74D
Integral of a power 0Q3HYFT9RQ
Darboux's theorem 0Q5A47J48D
The sequence r^n tends to zero for |r| < 1 0Q7K4AXE53
Irrationality of pi 0QHZX1PK8V
Ratio test 0QNY6ZBXPN
Convergent sequences are Cauchy 0QY87R9CY3
Continuity of the sum of a uniformly convergent series of functions 0R0MYT0BJD
pi is greater than 3 0R7SVNYTJS
Limit of a sum of sequences 0RAM4E07NR
Arccosine 0RDP2HEFEW
Cesàro means of a convergent sequence converge 0RDRPBMNAC
Logarithm is the inverse of the exponential 0RKW2X63YR
Exponential function 0RMVS7RXZX
A continuous nowhere differentiable function 0RXK2Z808Q
Power-reduction formula for cosine 0S049HG1W5
Abel's theorem 0S4EMDQS2F
Convergent sequences are bounded 0S4JF619V7
Indeterminate form 0S7G92KKA7
Derivative of tangent 0SAMR34WXC
Normal convergence 0T0EJDGHMB
Power series of arctangent 0T32ME0A6H
Dirichlet function 0T4EJ8ZJM9
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
Subsequences of a convergent sequence converge to the same limit 0VHAVXW08V
Term test 0VVY4H92G2
Continuous functions on a closed bounded interval are uniformly continuous 0W0TZDRF9C
Absolute convergence 0W3K3QPEWT
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
Uniform limit theorem 0X9XT05JVC
Wallis product 0XCYA4YERS
Squeeze theorem 0XMF00T4XQ
Dini's theorem 0XRRYF1HPE
Second fundamental theorem of calculus 0XYAFCN0PT
Absolutely convergent series converge 0Y123X48GA
Sequence of functions 0Y8FRYX8Q2
Term-by-term integration 0YBY3VYX0K
x^n converges pointwise but not uniformly on [0, 1] 0YSRR1JTKZ
Mercator series 0YZXNE3HFR
A function with nonnegative second derivative is convex 0Z41C17TGP
Constant multiple rule 0Z9JB71BS0
Cauchy criterion for sequences 0ZA4AX4MW3
p-series test 0ZNSSK8ZK1
The logarithm is strictly concave 0ZS54B9GJF
Uniqueness of limits 0ZXJ2455NJ

Sources

Web https://en.wikipedia.org/wiki/Limit_of_a_sequence, Real numbers
Web https://en.wikipedia.org/wiki/Limit_of_a_sequence, Topological spaces

What links here

Used by

Sequential characterization of limits 000MXHEY22
Limit of a quotient of sequences 03S6GWWKEZ
Pointwise convergence 056FSKG32J
Limit of a product of sequences 0734RSVF2V
Algebra of limits 07XBM3A9J5
The sequence 1/n tends to zero 08QAX4H1PF
Epsilon-N definition of the limit of a sequence 09G6KHS6GY
The nth root of n tends to 1 0AVEYZ2T0Q
Convergent sequence 0AZ3R52593
Monotone convergence theorem for sequences 0FM6N4R0VB
Limit comparison test 0HW7BNHHGF
Sequential continuity 0MVDQEAB60
Exponentials dominate powers in sequences 0MVTCFW42K
Limits preserve non-strict inequalities 0N32V0E3XJ
Stolz–Cesàro theorem 0Q3AKZV74D
The sequence r^n tends to zero for |r| < 1 0Q7K4AXE53
Limit of a sum of sequences 0RAM4E07NR
Cesàro means of a convergent sequence converge 0RDRPBMNAC
Subsequences of a convergent sequence converge to the same limit 0VHAVXW08V
Squeeze theorem 0XMF00T4XQ
Uniqueness of limits 0ZXJ2455NJ