Editing Limit of a sequence (section)

===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'''.