Editing Limit of a sequence (section)

==Sequence of more than one index==

Sometimes one may also consider a sequence with more than one index, for example, a double sequence <math>(x_{n, m})</math>. This sequence has a limit <math>L</math> if it becomes closer and closer to <math>L</math> when both ''n'' and ''m'' becomes very large.

===Example===

*If <math>x_{n, m} = c</math>  for constant <math display="inline">c</math>, then <math>x_{n,m} \to c</math>.
*If <math>x_{n, m} = \frac{1}{n + m}</math>, then <math>x_{n, m} \to 0</math>.
*If <math>x_{n, m} = \frac{n}{n + m}</math>, then the limit does not exist. Depending on the relative "growing speed" of <math display="inline">n</math> and <math display="inline">m</math>, this sequence can get closer to any value between <math display="inline">0</math> and <math display="inline">1</math>.

===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.

===Infinite limits===

A sequence <math>(x_{n,m})</math> is said to '''tend to infinity''', written
:<math>x_{n,m} \to \infty</math>, or 
:<math>\lim_{\begin{smallmatrix}
n \to \infty \\ m \to \infty
\end{smallmatrix}}x_{n,m} = \infty</math>,
if the following holds:
:For every real number <math>K</math>, there is 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} > K</math>; that is, the sequence terms are eventually larger than any fixed <math>K</math>.

Symbolically, this is:
:<math>\forall K \in \mathbb{R} \left(\exists N \in \N \left(\forall n, m \in \N \left(n, m \geq N \implies x_{n, m} > K \right)\right)\right)</math>.

Similarly, a sequence <math>(x_{n,m})</math> '''tends to minus infinity''', written
:<math>x_{n,m} \to -\infty</math>, or 
:<math>\lim_{\begin{smallmatrix}
n \to \infty \\ m \to \infty
\end{smallmatrix}}x_{n,m} = -\infty</math>,
if the following holds:
:For every real number <math>K</math>, there is 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} < K</math>; that is, the sequence terms are eventually smaller than any fixed <math>K</math>.

Symbolically, this is:
:<math>\forall K \in \mathbb{R} \left(\exists N \in \N \left(\forall n, m \in \N \left(n, m \geq N \implies x_{n, m} < K \right)\right)\right)</math>.

If a sequence tends to infinity or minus infinity, then it is divergent. However, a divergent sequence need not tend to plus or minus infinity, and the sequence <math>x_{n,m}=(-1)^{n+m}</math> provides one such example.

===Pointwise limits and uniform limits===

For a double sequence <math>(x_{n,m})</math>, we may take limit in one of the indices, say, <math>n \to \infty</math>, to obtain a single sequence <math>(y_m)</math>. In fact, there are two possible meanings when taking this limit. The first one is called '''pointwise limit''', denoted

:<math>x_{n, m} \to y_m\quad \text{pointwise}</math>, or
:<math>\lim_{n \to \infty} x_{n, m} = y_m\quad \text{pointwise}</math>,

which means:

:For each [[real number]] <math>\varepsilon > 0</math> and each fixed [[natural number]] <math>m</math>, there exists a natural number <math>N(\varepsilon, m) > 0</math> such that, for every natural number <math>n \geq N</math>, we have <math>|x_{n, m} - y_m| < \varepsilon</math>.<ref name="Habil">{{Cite web|url=https://www.researchgate.net/publication/242705642|date=2005|title=Double Sequences and Double Series|last=Habil|first=Eissa|language=en|access-date=2022-10-28}}</ref>

Symbolically, this is:
:<math>\forall \varepsilon > 0 \left( \forall m \in \mathbb{N} \left(\exists N \in \N \left(\forall n \in \N \left(n \geq N \implies |x_{n, m} - y_m| < \varepsilon \right)\right)\right)\right)</math>.

When such a limit exists, we say the sequence <math>(x_{n, m})</math> [[pointwise convergence|converges pointwise]] to <math>(y_m)</math>.

The second one is called '''uniform limit''', denoted

:<math>x_{n, m} \to y_m \quad \text{uniformly}</math>,
:<math>\lim_{n \to \infty} x_{n, m} = y_m \quad \text{uniformly}</math>,
:<math>x_{n, m} \rightrightarrows y_m </math>, or
:<math>\underset{n\to\infty}{\mathrm{unif} \lim} \; x_{n, m} = y_m </math>,

which means:

:For each [[real number]] <math>\varepsilon > 0</math>, there exists a natural number <math>N(\varepsilon) > 0</math> such that, for every [[natural number]] <math>m</math> and for every natural number <math>n \geq N</math>, we have <math>|x_{n, m} - y_m| < \varepsilon</math>.<ref name="Habil"/>

Symbolically, this is:
:<math>\forall \varepsilon > 0 \left(\exists N \in \N \left( \forall m \in \mathbb{N} \left(\forall n \in \N \left(n \geq N \implies |x_{n, m} - y_m| < \varepsilon \right)\right)\right)\right)</math>.

In this definition, the choice of <math>N</math> is independent of <math>m</math>. In other words, the choice of <math>N</math> is ''uniformly applicable'' to all natural numbers <math>m</math>. Hence, one can easily see that uniform convergence is a stronger property than pointwise convergence: the existence of uniform limit implies the existence and equality of pointwise limit:

:If <math>x_{n, m} \to y_m</math> uniformly, then <math>x_{n, m} \to y_m</math> pointwise.

When such a limit exists, we say the sequence <math>(x_{n, m})</math> [[uniform convergence|converges uniformly]] to <math>(y_m)</math>.

===Iterated limit===

For a double sequence <math>(x_{n,m})</math>, we may take limit in one of the indices, say, <math>n \to \infty</math>, to obtain a single sequence <math>(y_m)</math>, and then take limit in the other index, namely <math>m \to \infty</math>, to get a number <math>y</math>. Symbolically,
:<math>\lim_{m \to \infty} \lim_{n \to \infty} x_{n, m} = \lim_{m \to \infty} y_m = y</math>.

This limit is known as '''[[iterated limit]]''' of the double sequence. The order of taking limits may affect the result, i.e.,

:<math>\lim_{m \to \infty} \lim_{n \to \infty} x_{n, m} \ne \lim_{n \to \infty} \lim_{m \to \infty} x_{n, m}</math> in general.

A sufficient condition of equality is given by the [[Moore-Osgood theorem]], which requires the limit <math>\lim_{n \to \infty}x_{n, m} = y_m</math> to be uniform in <math display="inline">m</math>.<ref name="Zakon" />