Editing Limit of a sequence (section)

===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>.