Editing Limit of a sequence (section)

===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>).