Editing Limit of a sequence (section)

==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 <math display="inline">c</math>, 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 an integer <math>N > \frac{1}{\varepsilon}.</math> For every <math>n \geq N</math>, one has <math>|x_n - 0| =\frac 1n \le \frac 1N  < \varepsilon</math>.</ref><ref name=":0" />
*If <math>x_n = \frac{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 display="inline">0.3, 0.33, 0.333, 0.3333, \dots</math> converges to <math display="inline">\frac{1}{3}</math>. The [[decimal representation]] <math display="inline">0.3333\dots</math> is the ''limit'' of the previous sequence, defined by <math display="block"> 0.3333... : = \lim_{n\to\infty} \sum_{k=1}^n \frac{3}{10^k}</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.

===Definition===
We call <math>x</math> the '''limit''' of the [[sequence]] <math>(x_n)</math>, which is written
:<math>x_n \to x</math>, or 
:<math>\lim_{n\to\infty} x_n = x</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>.

Symbolically, this is:
:<math>\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>|alt=Example of a sequence which converges to the limit                 <nowiki> </nowiki>       a             <nowiki> </nowiki>   {\displaystyle a}    .
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>\varepsilon_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===
Some other important properties of limits of real sequences include the following: 
*When it exists, the limit of a sequence is unique.<ref name=":0" />
*Limits of sequences behave well with respect to the usual [[Arithmetic#Arithmetic operations|arithmetic operations]]. If <math>\lim_{n\to\infty} a_n</math> and <math>\lim_{n\to\infty} b_n</math> exists, then
::<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) =  \left(\lim_{n\to\infty} a_n \right)\cdot \left( \lim_{n\to\infty} b_n \right)</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>

*For any [[continuous function]] <math display="inline">f</math>, if <math>\lim_{n\to\infty}x_n</math> exists, then <math>\lim_{n\to\infty} f \left(x_n \right)</math> exists too. In fact, any real-valued [[function (mathematics)|function]] ''<math display="inline">f</math>'' 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).

*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</math> greater than some <math>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>.
*([[Monotone convergence theorem]]) If <math>a_n</math> is [[Sequence#Bounded|bounded]] and [[Sequence#Increasing and decreasing|monotonic]] for all <math>n</math> greater than some <math>N</math>, 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 the following holds:
:For every real number <math>K</math>, there is a natural number <math>N</math> such that for every natural number <math>n \geq N</math>, we have <math>x_n > K</math>; that is, the sequence terms are eventually larger than any fixed <math>K</math>.

Symbolically, this is:
:<math>\forall K \in \mathbb{R} \left(\exists N \in \N \left(\forall n \in \N \left(n \geq N \implies x_n > K \right)\right)\right)</math>.

Similarly, we say a sequence '''tends to minus infinity''', written
:<math>x_n \to -\infty</math>, or
:<math>\lim_{n\to\infty}x_n = -\infty</math>,
if the following holds:
:For every real number <math>K</math>, there is a natural number <math>N</math> such that for every natural number <math>n \geq N</math>, we have <math>x_n < K</math>; that is, the sequence terms are eventually smaller than any fixed <math>K</math>.

Symbolically, this is:
:<math>\forall K \in \mathbb{R} \left(\exists N \in \N \left(\forall n \in \N \left(n \geq N \implies x_n < K \right)\right)\right)</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.