Editing Limit of a sequence (section)

===Definition===
We call <math>x</math> the '''double limit''' of the [[sequence]] <math>(x_{n, m})</math>, written
:<math>x_{n, m} \to x</math>, or 
:<math>\lim_{\begin{smallmatrix}
n \to \infty \\ m \to \infty
\end{smallmatrix}} x_{n, m} = 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 pair of natural numbers <math>n, m \geq N</math>, we have <math>|x_{n, m} - x| < \varepsilon</math>.<ref name="Zakon">{{cite book|chapter=Chapter 4. Function Limits and Continuity|pages=223|title=Mathematical Anaylysis, Volume I|year=2011|last1=Zakon|first1=Elias|publisher=University of Windsor |isbn=9781617386473}}</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, m})</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, m \in \N \left(n, m \geq N \implies |x_{n, m} - x| < \varepsilon \right)\right)\right) </math>.

The double limit is different from taking limit in ''n'' first, and then in ''m''. The latter is known as [[iterated limit]]. Given that both the double limit and the iterated limit exists, they have the same value. However, it is possible that one of them exist but the other does not.