Editing Limit of a sequence (section)

===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.