Sequences:Limits - Revision history

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Lila: Created page with "In mathematics, the '''limit of a sequence''' is the value that the terms of a sequence "tend to", and is often denoted using the $\lim$ symbol (e.g.,
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Created page with "In mathematics, the '''limit of a sequence''' is the value that the terms of a sequence "tend to", and is often denoted using the <math>\lim</math> symbol (e.g., <math..."

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In [[mathematics]], the '''limit of a sequence''' is the value that the terms of a [[sequence]] "tend to", and is often denoted using the <math>\lim</math> symbol (e.g., <math>\lim_{n \to \infty}a_n</math>).<ref name="Courant (1961), p. 29">Courant (1961), p. 29.</ref> If such a limit exists, the sequence is called '''convergent'''.<ref>{{Cite web|last=Weisstein|first=Eric W.|title=Convergent Sequence|url=https://mathworld.wolfram.com/ConvergentSequence.html|access-date=2020-08-18|website=mathworld.wolfram.com|language=en}}</ref> A sequence that does not converge is said to be '''divergent'''.<ref>Courant (1961), p. 39.</ref> The limit of a sequence is said to be the fundamental notion on which the whole of [[mathematical analysis]] ultimately rests.<ref name="Courant (1961), p. 29"/>

Limits can be defined in any [[metric space|metric]] or [[topological space]], but are usually first encountered in the [[real number]]s.

==Real numbers==
[[File:Converging Sequence example.svg|320px|thumb|The plot of a convergent sequence {''a<sub>n</sub>''} is shown in blue. Here, one can see that the sequence is converging to the limit 0 as ''n'' increases.]]

In the [[real numbers]], a number <math>L</math> is the '''limit''' of the [[sequence]] <math>(x_n),</math> if the numbers in the sequence become closer and closer to <math>L</math>—and not to any other number.

===Examples===
{{see also|List of limits}}
If <math>x_n = c</math> for constant ''c'', then <math>x_n \to c.</math><ref group="proof">''Proof'': choose <math>N = 1.</math> For every <math>n \geq N,</math> <math>|x_n - c| = 0 < \varepsilon</math></ref><ref name=":0">{{Cite web|title=Limits of Sequences {{!}} Brilliant Math & Science Wiki|url=https://brilliant.org/wiki/limits-of-sequences/|access-date=2020-08-18|website=brilliant.org|language=en-us}}</ref>
If <math>x_n = \frac{1}{n},</math> then <math>x_n \to 0.</math><ref group="proof">''Proof'': choose <math>N = \left\lfloor\frac{1}{\varepsilon}\right\rfloor + 1</math> (the [[Floor and ceiling functions|floor function]]). For every <math>n \geq N,</math> <math>|x_n - 0| \le x_N = \frac{1}{\lfloor 1/\varepsilon \rfloor + 1} < \varepsilon.</math></ref><ref name=":0" />
If <math>x_n = 1/n</math> when <math>n</math> is even, and <math>x_n = \frac{1}{n^2}</math> when <math>n</math> is odd, then <math>x_n \to 0.</math> (The fact that <math>x_{n+1} > x_n</math> whenever <math>n</math> is odd is irrelevant.)
Given any real number, one may easily construct a sequence that converges to that number by taking decimal approximations. For example, the sequence <math>0.3, 0.33, 0.333, 0.3333, \dots</math> converges to <math>1/3.</math> Note that the [[decimal representation]] <math>0.3333...</math> is the ''limit'' of the previous sequence, defined by <math display="block"> 0.3333... : = \lim_{n\to\infty} \sum_{i=1}^n \frac{3}{10^i}.</math>
Finding the limit of a sequence is not always obvious. Two examples are <math>\lim_{n\to\infty} \left(1 + \tfrac{1}{n}\right)^n</math> (the limit of which is the [[e (mathematical constant)|number ''e'']]) and the [[Arithmetic–geometric mean]]. The [[squeeze theorem]] is often useful in the establishment of such limits.

===Formal definition===
We call <math>x</math> the '''limit''' of the [[sequence]] <math>(x_n)</math> if the following condition holds:
For each [[real number]] <math>\varepsilon > 0,</math> there exists a [[natural number]] <math>N</math> such that, for every natural number <math>n \geq N,</math> we have <math>|x_n - x| < \varepsilon.</math><ref>{{Cite web| last=Weisstein|first=Eric W.| title=Limit|url=https://mathworld.wolfram.com/Limit.html|access-date=2020-08-18| website=mathworld.wolfram.com|language=en}}</ref>
In other words, for every measure of closeness <math>\varepsilon,</math> the sequence's terms are eventually that close to the limit. The sequence <math>(x_n)</math> is said to '''converge to''' or '''tend to''' the limit <math>x,</math> written <math>x_n \to x</math> or <math>\lim_{n\to\infty} x_n = x.</math>

Symbolically, this is:
<math display="block">\forall \varepsilon > 0 \left(\exists N \in \N \left(\forall n \in \N \left(n \geq N \implies |x_n - x| < \varepsilon \right)\right)\right). </math>

{{anchor|null sequence}}
If a sequence <math>(x_n)</math> converges to some limit <math>x,</math> then it is '''convergent''' and <math>x</math> is the only limit; otherwise <math>(x_n)</math> is '''divergent'''. A sequence that has zero as its limit is sometimes called a '''null sequence'''.

=== Illustration ===
<gallery widths="350" heights="200">
File:Folgenglieder im KOSY.svg|Example of a sequence which converges to the limit <math>a.</math>
File:Epsilonschlauch.svg|Regardless which <math>\varepsilon > 0</math> we have, there is an index <math>N_0,</math> so that the sequence lies afterwards completely in the epsilon tube <math>(a-\varepsilon,a+\varepsilon).</math>
File:Epsilonschlauch klein.svg|There is also for a smaller <math>\epsilon_1 > 0</math> an index <math>N_1,</math> so that the sequence is afterwards inside the epsilon tube <math>(a-\varepsilon_1,a+\varepsilon_1).</math>
File:Epsilonschlauch2.svg|For each <math>\varepsilon > 0</math> there are only finitely many sequence members outside the epsilon tube.
</gallery>

===Properties===
Limits of sequences behave well with respect to the usual [[Arithmetic#Arithmetic operations|arithmetic operations]]. If <math>a_n \to a</math> and <math>b_n \to b,</math> then <math>a_n+b_n \to a+b,</math> <math>a_n\cdot b_n \to ab</math> and, if neither ''b'' nor any <math>b_n</math> is zero, <math>\frac{a_n}{b_n} \to \frac{a}{b}.</math><ref name=":0" />

For any [[continuous function]] ''f'', if <math>x_n \to x</math> then <math>f(x_n) \to f(x).</math> In fact, any real-valued [[function (mathematics)|function]] ''f'' is continuous if and only if it preserves the limits of sequences (though this is not necessarily true when using more general notions of continuity).

Some other important properties of limits of real sequences include the following (provided, in each equation below, that the limits on the right exist).

The limit of a sequence is unique.<ref name=":0" />
<math>\lim_{n\to\infty} (a_n \pm b_n) = \lim_{n\to\infty} a_n \pm \lim_{n\to\infty} b_n</math><ref name=":0" />
<math>\lim_{n\to\infty} c a_n = c \cdot \lim_{n\to\infty} a_n</math><ref name=":0" />
<math>\lim_{n\to\infty} (a_n \cdot b_n) = (\lim_{n\to\infty} a_n)\cdot( \lim_{n\to\infty} b_n)</math><ref name=":0" />
<math>\lim_{n\to\infty} \left(\frac{a_n}{b_n}\right) = \frac{\lim\limits_{n\to\infty} a_n}{\lim\limits_{n\to\infty} b_n}</math> provided <math>\lim_{n\to\infty} b_n \ne 0</math><ref name=":0" />
<math>\lim_{n\to\infty} a_n^p = \left[ \lim_{n\to\infty} a_n \right]^p</math>
If <math>a_n \leq b_n</math> for all <math>n</math> greater than some <math>N,</math> then <math>\lim_{n\to\infty} a_n \leq \lim_{n\to\infty} b_n .</math>
([[Squeeze theorem]]) If <math>a_n \leq c_n \leq b_n</math> for all <math>n > N,</math> and <math>\lim_{n\to\infty} a_n = \lim_{n\to\infty} b_n = L,</math> then <math>\lim_{n\to\infty} c_n = L.</math>
If a sequence is [[Sequence#Bounded|bounded]] and [[Sequence#Increasing and decreasing|monotonic]], then it is convergent.
A sequence is convergent if and only if every subsequence is convergent.
*If every subsequence of a sequence has its own subsequence which converges to the same point, then the original sequence converges to that point.

These properties are extensively used to prove limits, without the need to directly use the cumbersome formal definition. For example. once it is proven that <math>1/n \to 0,</math> it becomes easy to show—using the properties above—that <math>\frac{a}{b+\frac{c}{n}} \to \frac{a}{b}</math> (assuming that <math>b \ne 0</math>).

===Infinite limits===

A sequence <math>(x_n)</math> is said to '''tend to infinity''', written <math>x_n \to \infty</math> or <math>\lim_{n\to\infty}x_n = \infty,</math> if for every ''K'', there is an ''N'' such that for every <math>n \geq N,</math> <math>x_n > K</math>; that is, the sequence terms are eventually larger than any fixed ''K''.

Similarly, <math>x_n \to -\infty</math> if for every ''K'', there is an ''N'' such that for every <math>n \geq N,</math> <math>x_n < K.</math> If a sequence tends to infinity or minus infinity, then it is divergent. However, a divergent sequence need not tend to plus or minus infinity, and the sequence <math>x_n=(-1)^n</math> provides one such example.

==Metric spaces==

===Definition===

A point <math>x</math> of the [[metric space]] <math>(X, d)</math> is the '''limit''' of the [[sequence]] <math>(x_n)</math> if for all <math>\epsilon > 0,</math> there is an <math>N</math> such that, for every <math>n \geq N,</math> <math>d(x_n, x) < \epsilon.</math> This coincides with the definition given for real numbers when <math>X = \R</math> and <math>d(x, y) = |x-y|.</math>

===Properties===

For any [[continuous function]] ''f'', if <math>x_n \to x</math> then <math>f(x_n) \to f(x).</math> In fact, a [[Function (mathematics)|function]] ''f'' is continuous if and only if it preserves the limits of sequences.

Limits of sequences are unique when they exist, as distinct points are separated by some positive distance, so for <math>\epsilon</math> less than half this distance, sequence terms cannot be within a distance <math>\epsilon</math> of both points.

==Topological spaces==

===Definition===

A point <math>x \in X</math> of the topological space <math>(X, \tau)</math> is a '''{{visible anchor|Limit of a sequence in a topological space|text=limit}}''' or '''{{visible anchor|Limit point of a sequence|text=limit point}}'''{{sfn|Dugundji|1966|pp=209-210}}{{sfn|Császár|1978|p=61}} of the [[sequence]] <math>\left(x_n\right)_{n \in \N}</math> if for every [[Topological neighbourhood|neighbourhood]] <math>U</math> of <math>x,</math> there exists some <math>N \in \N</math> such that for every <math>n \geq N,</math> <math>x_n \in U.</math><ref>{{cite book|last1=Zeidler|first1=Eberhard|title=Applied functional analysis : main principles and their applications|date=1995|publisher=Springer-Verlag|location=New York|isbn=978-0-387-94422-7|page=29|edition=1}}</ref> This coincides with the definition given for metric spaces, if <math>(X, d)</math> is a metric space and <math>\tau</math> is the topology generated by <math>d.</math>

A limit of a sequence of points <math>\left(x_n\right)_{n \in \N}</math> in a topological space <math>T</math> is a special case of a [[Limit of a function#Functions on topological spaces|limit of a function]]: the [[Domain of a function|domain]] is <math>\N</math> in the space <math>\N \cup \lbrace + \infty \rbrace,</math> with the [[induced topology]] of the [[affinely extended real number system]], the [[Range of a function|range]] is <math>T,</math> and the function argument <math>n</math> tends to <math>+\infty,</math> which in this space is a [[limit point]] of <math>\N.</math>

===Properties===

In a [[Hausdorff space]], limits of sequences are unique whenever they exist. Note that this need not be the case in non-Hausdorff spaces; in particular, if two points <math>x</math> and <math>y</math> are [[topologically indistinguishable]], then any sequence that converges to <math>x</math> must converge to <math>y</math> and vice versa.

@@ Line 116: / Line 116: @@
 In a Hausdorff space, limits of sequences are unique whenever they exist. Note that this need not be the case in non-Hausdorff spaces; in particular, if two points <math>x</math> and <math>y</math> are topologically indistinguishable, then any sequence that converges to <math>x</math> must converge to <math>y</math> and vice versa.
-==References==
+== Licensing ==
-# Császár, Ákos (1978). General topology. Translated by Császár, Klára. Bristol England: Adam Hilger Ltd. ISBN 0-85274-275-4. OCLC 4146011.
+Content obtained and/or adapted from:
-# Dugundji, James (1966). Topology. Boston: Allyn and Bacon. ISBN 978-0-697-06889-7. OCLC 395340485.
+* [https://en.wikipedia.org/wiki/Limit_of_a_sequence Limit of a sequence, Wikipedia] under a CC BY-SA license

@@ Line 72: / Line 72: @@
 Some other important properties of limits of real sequences include the following (provided, in each equation below, that the limits on the right exist).
-*The limit of a sequence is unique.<ref name=":0" />
+*The limit of a sequence is unique.
 *<math>\lim_{n\to\infty} (a_n \pm b_n) =  \lim_{n\to\infty} a_n \pm \lim_{n\to\infty} b_n</math>
 *<math>\lim_{n\to\infty} c a_n =  c \cdot \lim_{n\to\infty} a_n</math>

@@ Line 73: / Line 73: @@
 *The limit of a sequence is unique.<ref name=":0" />
-*<math>\lim_{n\to\infty} (a_n \pm b_n) =  \lim_{n\to\infty} a_n \pm \lim_{n\to\infty} b_n</math><ref name=":0" />
+*<math>\lim_{n\to\infty} (a_n \pm b_n) =  \lim_{n\to\infty} a_n \pm \lim_{n\to\infty} b_n</math>
-*<math>\lim_{n\to\infty} c a_n =  c \cdot \lim_{n\to\infty} a_n</math><ref name=":0" />
+*<math>\lim_{n\to\infty} c a_n =  c \cdot \lim_{n\to\infty} a_n</math>
-*<math>\lim_{n\to\infty} (a_n \cdot b_n) =  (\lim_{n\to\infty} a_n)\cdot( \lim_{n\to\infty} b_n)</math><ref name=":0" />
+*<math>\lim_{n\to\infty} (a_n \cdot b_n) =  (\lim_{n\to\infty} a_n)\cdot( \lim_{n\to\infty} b_n)</math>
-*<math>\lim_{n\to\infty} \left(\frac{a_n}{b_n}\right) = \frac{\lim\limits_{n\to\infty} a_n}{\lim\limits_{n\to\infty} b_n}</math> provided <math>\lim_{n\to\infty} b_n \ne 0</math><ref name=":0" />
+*<math>\lim_{n\to\infty} \left(\frac{a_n}{b_n}\right) = \frac{\lim\limits_{n\to\infty} a_n}{\lim\limits_{n\to\infty} b_n}</math> provided <math>\lim_{n\to\infty} b_n \ne 0</math>
 *<math>\lim_{n\to\infty} a_n^p =  \left[ \lim_{n\to\infty} a_n \right]^p</math>
 *If <math>a_n \leq b_n</math> for all <math>n</math> greater than some <math>N,</math> then <math>\lim_{n\to\infty} a_n \leq \lim_{n\to\infty} b_n .</math>
@@ Line 115: / Line 115: @@
 In a Hausdorff space, limits of sequences are unique whenever they exist. Note that this need not be the case in non-Hausdorff spaces; in particular, if two points <math>x</math> and <math>y</math> are topologically indistinguishable, then any sequence that converges to <math>x</math> must converge to <math>y</math> and vice versa.

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		+	[[File:Archimedes pi.svg\|350px\|right\|thumb\|alt=diagram of a hexagon and pentagon circumscribed outside a circle\|The sequence given by the perimeters of regular ''n''-sided [[polygon]]s that circumscribe the [[unit circle]] has a limit equal to the perimeter of the circle, i.e. <math>2\pi r.</math> The corresponding sequence for inscribed polygons has the same limit.]]
		+	<div class="thumb tright">
		+	<div class="thumbinner" style="width:252px;">
		+	<div style="width:240px; font-family:arial; font-size:12px; font-weight:bold; background:#fff;">
		+	{\| class="wikitable" style="width:100%;"
		+	\|-
		+	!''n''!!''n'' sin(1/''n'')
		+	\|-
		+	\|1\|\|0.841471
		+	\|-
		+	\|2\|\|0.958851
		+	\|-
		+	\|colspan="2"\|...
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		+	\|10\|\|0.998334
		+	\|-
		+	\|colspan="2"\|...
		+	\|-
		+	\|100\|\|0.999983
		+	\|}
		+	</div>
		+	<div class="thumbcaption">
		+	As the positive [[integer]] <math>n</math> becomes larger and larger, the value <math>n\cdot \sin\left(\tfrac1{n}\right)</math> becomes arbitrarily close to <math>1.</math> We say that "the limit of the sequence <math>n\cdot \sin\left(\tfrac1{n}\right)</math> equals <math>1.</math>"
		+	</div>
		+	</div>
		+	</div>
		+
	In [[mathematics]], the '''limit of a sequence''' is the value that the terms of a [[sequence]] "tend to", and is often denoted using the <math>\lim</math> symbol (e.g., <math>\lim_{n \to \infty}a_n</math>).<ref name="Courant (1961), p. 29">Courant (1961), p. 29.</ref> If such a limit exists, the sequence is called '''convergent'''.<ref>{{Cite web\|last=Weisstein\|first=Eric W.\|title=Convergent Sequence\|url=https://mathworld.wolfram.com/ConvergentSequence.html\|access-date=2020-08-18\|website=mathworld.wolfram.com\|language=en}}</ref> A sequence that does not converge is said to be '''divergent'''.<ref>Courant (1961), p. 39.</ref> The limit of a sequence is said to be the fundamental notion on which the whole of [[mathematical analysis]] ultimately rests.<ref name="Courant (1961), p. 29"/>		In [[mathematics]], the '''limit of a sequence''' is the value that the terms of a [[sequence]] "tend to", and is often denoted using the <math>\lim</math> symbol (e.g., <math>\lim_{n \to \infty}a_n</math>).<ref name="Courant (1961), p. 29">Courant (1961), p. 29.</ref> If such a limit exists, the sequence is called '''convergent'''.<ref>{{Cite web\|last=Weisstein\|first=Eric W.\|title=Convergent Sequence\|url=https://mathworld.wolfram.com/ConvergentSequence.html\|access-date=2020-08-18\|website=mathworld.wolfram.com\|language=en}}</ref> A sequence that does not converge is said to be '''divergent'''.<ref>Courant (1961), p. 39.</ref> The limit of a sequence is said to be the fundamental notion on which the whole of [[mathematical analysis]] ultimately rests.<ref name="Courant (1961), p. 29"/>