Definition:Convergence
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[edit] Convergent Sequence
[edit] Topological Space
Let
be a topological space.
Let
be a sequence in
.
Then
converges to the limit
if:
- for any open set
such that
:
.
This can be alternatively stated:
converges to the limit
if:
- every open set in
containing
contains all but a finite number of terms of
.
Such a sequence is convergent.
[edit] Metric Space
Let
be a metric space.
Let
be a sequence in
.
Then
converges to the limit
iff:
Or equivalently, using the definition of neighborhood:
We can write "
as
", or
.
This is voiced "As
tends to infinity,
tends to (the limit)
."
It can be seen that by the definition of open set in a metric space, this definition is equivalent to that for convergence in a topological space.
[edit] Standard Number Fields
When
is one of the standard number fields
, and the metric
is the usual (Euclidean) metric, the condition on convergence becomes:
converges to the limit
iff:
where
is the modulus of
.
The validity of this definition derives from the fact that:
- the Rational Numbers form Metric Space;
- the Real Number Line is Metric Space;
- the Complex Plane is Metric Space.
[edit] Convergent Series
Let
be one of the standard number fields
.
Let
be a series in
.
Let
be the sequence of partial sums of
.
It follows that
can be treated as a sequence in the metric space
.
If
as
, the series converges to the sum
.
[edit] Convergent Function
[edit] Convergence of a Function on a Metric Space
Let
and
be metric spaces.
Let
be a limit point of
.
Let
be a mapping from
to
defined everywhere on
except possibly at
.
Let
tend to the limit
as
tends to
.
Then
converges to the limit
as
tends to
.
[edit] Convergence of Real and Complex Functions
As:
- The real number line
under the usual metric forms a metric space;
- The complex plane
under the usual metric forms a metric space;
the definition holds for real and complex functions.
[edit] Convergent Filter
Let
be a topological space.
Let
be a filter on
.
Then
converges to a point
if:
where
is a neighborhood of
.
That is, a filter is convergent to a point
if every neighborhood of
is an element of that filter.

