# Definition:Convergence

## Convergent Sequence

### Topological Space

Let $T = \struct {S, \tau}$ be a topological space.

Let $\sequence {x_n}_{n \mathop \in \N}$ be an infinite sequence in $S$.

Then $\sequence {x_n}$ converges to the limit $\alpha \in S$ if and only if:

$\forall U \in \tau: \alpha \in U \implies \paren {\exists N \in \R_{>0}: \forall n \in \N: n > N \implies x_n \in U}$

### Metric Space

Let $M = \left({A, d}\right)$ be a metric space or a pseudometric space.

Let $\sequence {x_k}$ be a sequence in $A$.

$\sequence {x_k}$ converges to the limit $l \in A$ if and only if:

$\forall \epsilon \in \R_{>0}: \exists N \in \R_{>0}: \forall n \in \N: n > N \implies \map d {x_n, l} < \epsilon$

For other equivalent definitions of a convergent sequence in a Metric Space see: Definition:Convergent Sequence in Metric Space

### Normed Division Ring

Let $\struct {R, \norm {\, \cdot \,} }$ be a normed division ring.

Let $\sequence {x_n}$ be a sequence in $R$.

The sequence $\sequence {x_n}$ converges to $x \in R$ in the norm $\norm {\, \cdot \,}$ if and only if:

$\forall \epsilon \in \R_{>0}: \exists N \in \R_{>0}: \forall n \in \N: n > N \implies \norm {x_n - x} < \epsilon$

### Real Numbers

Let $\sequence {x_k}$ be a sequence in $\R$.

The sequence $\sequence {x_k}$ converges to the limit $l \in \R$ if and only if:

$\forall \epsilon \in \R_{>0}: \exists N \in \R_{>0}: n > N \implies \size {x_n - l} < \epsilon$

where $\size x$ denotes the absolute value of $x$.

### Rational Numbers

Let $\sequence {x_k}$ be a sequence in $\Q$.

$\sequence {x_k}$ converges to the limit $l \in \R$ if and only if:

$\forall \epsilon \in \R_{>0}: \exists N \in \R_{>0}: n > N \implies \size {x_n - l} < \epsilon$

where $\size x$ is the absolute value of $x$.

### Complex Numbers

Let $\sequence {z_k}$ be a sequence in $\C$.

$\sequence {z_k}$ converges to the limit $c \in \C$ if and only if:

$\forall \epsilon \in \R_{>0}: \exists N \in \R: n > N \implies \cmod {z_n - c} < \epsilon$

where $\cmod z$ denotes the modulus of $z$.

## Convergent Series

Let $\left({S, \circ, \tau}\right)$ be a topological semigroup.

Let $\displaystyle \sum_{n \mathop = 1}^\infty a_n$ be a series in $S$.

This series is said to be convergent if and only if its sequence of partial sums $\left \langle {s_N} \right \rangle$ converges in the topological space $\left({S, \tau}\right)$.

If $s_N \to s$ as $N \to \infty$, the series converges to the sum $s$, and one writes $\displaystyle \sum_{n \mathop = 1}^\infty a_n = s$.

### Convergent Series in a Normed Vector Space (Definition 1)

Let $V$ be a normed vector space.

Let $d$ be the induced metric on $V$.

Let $\displaystyle S := \sum_{n \mathop = 1}^\infty a_n$ be a series in $V$.

$S$ is convergent if and only if its sequence $\left \langle {s_N} \right \rangle$ of partial sums converges in the metric space $\left({V, d}\right)$.

### Convergent Series in a Normed Vector Space (Definition 2)

Let $\struct {V, \norm {\, \cdot \,}}$ be a normed vector space.

Let $\displaystyle S := \sum_{n \mathop = 1}^\infty a_n$ be a series in $V$.

$S$ is convergent if and only if its sequence $\sequence {s_N}$ of partial sums converges in the normed vector space $\struct {V, \norm {\, \cdot \,}}$.

### Convergent Series in a Number Field

Let $S$ be one of the standard number fields $\Q, \R, \C$.

Let $\displaystyle \sum_{n \mathop = 1}^\infty a_n$ be a series in $S$.

Let $\sequence {s_N}$ be the sequence of partial sums of $\displaystyle \sum_{n \mathop = 1}^\infty a_n$.

It follows that $\sequence {s_N}$ can be treated as a sequence in the metric space $S$.

If $s_N \to s$ as $N \to \infty$, the series converges to the sum $s$, and one writes $\displaystyle \sum_{n \mathop = 1}^\infty a_n = s$.

A series is said to be convergent if and only if it converges to some $s$.

## Convergent Mapping

### Metric Space

Let $M_1 = \left({A_1, d_1}\right)$ and $M_2 = \left({A_2, d_2}\right)$ be metric spaces.

Let $c$ be a limit point of $M_1$.

Let $f: A_1 \to A_2$ be a mapping from $A_1$ to $A_2$ defined everywhere on $A_1$ except possibly at $c$.

Let $f \left({x}\right)$ tend to the limit $L$ as $x$ tends to $c$.

Then $f$ converges to the limit $L$ as $x$ tends to $c$.

### Real Function

As the real number line $\R$ under the usual (Euclidean) metric forms a metric space, the definition also holds for real functions:

Let $f: \R \to \R$ be a real function defined everywhere on $A_1$ except possibly at $c$.

Let $\map f x$ tend to the limit $L$ as $x$ tends to $c$.

Then $f$ converges to the limit $L$ as $x$ tends to $c$.

### Complex Function

As the complex plane $\C$ under the usual (Euclidean) metric forms a metric space, the definition also holds for complex functions:

Let $f: \C \to \C$ be a complex function defined everywhere on $\C$ except possibly at $c$.

Let $f \left({z}\right)$ tend to the limit $L$ as $z$ tends to $c$.

Then $f$ converges to the limit $L$ as $z$ tends to $c$.

## Convergent Filter

Let $\struct {S, \tau}$ be a topological space.

Let $\FF$ be a filter on $S$.

Then $\FF$ converges to a point $x \in S$ if and only if:

$\forall N_x \subseteq S: N_x \in \FF$

where $N_x$ is a neighborhood of $x$.

That is, a filter converges to a point $x$ if and only if every neighborhood of $x$ is an element of that filter.

If there is a point $x \in S$ such that $\FF$ converges to $x$, then $\FF$ is convergent.

## Convergent Filter Basis

Let $\struct {S, \tau}$ be a topological space.

Let $\BB$ be a filter basis of a filter $\FF$ on $S$.

Then $\BB$ converges to a point $x \in S$ if and only if:

$\forall N_x \subseteq S: \exists B \in \BB: B \subseteq N_x$

where $N_x$ is a neighborhood of $x$.

That is, a filter basis is convergent to a point $x$ if every neighborhood of $x$ contains some set of that filter basis.