Editing Sequence (section)

===Applications and important results===
If <math>(a_n)</math> and <math>(b_n)</math> are convergent sequences, then the following limits exist, and can be computed as follows:<ref name="Gaughan" /><ref name="Dawkins">{{cite web |url=http://tutorial.math.lamar.edu/Classes/CalcII/Sequences.aspx |title=Series and Sequences |last1=Dawikins |first1=Paul |work=Paul's Online Math Notes/Calc II (notes) |access-date=18 December 2012 |archive-date=30 November 2012 |archive-url=https://web.archive.org/web/20121130095834/http://tutorial.math.lamar.edu/Classes/CalcII/Sequences.aspx |url-status=live }}</ref>

* <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 \lim_{n\to\infty} a_n</math> for all real numbers <math>c</math>
* <math>\lim_{n\to\infty} (a_n b_n) = \bigl( \lim_{n\to\infty} a_n \bigr) \bigl( \lim_{n\to\infty} b_n \bigr)</math>
* <math>\lim_{n\to\infty} \frac{a_n} {b_n} = \bigl( \lim \limits_{n\to\infty} a_n \bigr) \big/ \bigl( \lim \limits_{n\to\infty} b_n \bigr)</math>, provided that <math>\lim_{n\to\infty} b_n \ne 0</math>
* <math>\lim_{n\to\infty} a_n^p = \bigl( \lim_{n\to\infty} a_n \bigr)^p</math> for all <math>p > 0</math> and <math>a_n > 0</math>

Moreover:
* 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>.{{efn|If the inequalities are replaced by strict inequalities then this is false: There are sequences such that <math>a_n < b_n</math> for all <math>n</math>, but <math>\lim_{n\to\infty} a_n = \lim_{n\to\infty} b_n </math>.}}
* ([[Squeeze Theorem]])<br>If <math>(c_n)</math> is a sequence such that <math>a_n \leq c_n \leq b_n</math> for all <math>n > N</math> {{nowrap|and <math>\lim_{n\to\infty} a_n = \lim_{n\to\infty} b_n = L</math>,}}<br>then <math>(c_n)</math> is convergent, and <math>\lim_{n\to\infty} c_n = L</math>.
* If a sequence is [[#Bounded|bounded]] and [[#Increasing and decreasing|monotonic]] then it is convergent.
* A sequence is convergent if and only if all of its subsequences are convergent.