Convergence:
Definition
First, Lets translate the calculus definition of convergence, to the "language" of metric spaces: We say that a sequence
converges to
if for every
exists
that for each
the following holds:
.
Equivalently, we can define converges using Open-balls: A sequence
converges to
If for every
exists
that for each
the following holds:
.
Equivalently, we can define converges using Open-balls: A sequence
The latter definition uses the "language" of open-balls, But we can do better - We can remove the
from the definition of convergence, thus making the definition more topological. Let's define that
converges to
(and mark
) , if for every ball
around
, exists
that for each
the following holds:
.
is called the limit of the sequence.
The definitions are all the same, but the latter uses topological terms, and can be easily converted to a topological definition later.
Properties
- If a sequence has a limit, it has only one limit.
Proof Let a sequencehave two limits,
and
. If they are not the same, we must have
. Let
be smaller than this distance. Now for some
, for all
, it must be the case that both
and
by virtue of the fact
and
are limits. But this is impossible; the two balls are separate. Therefore the limits are coincident, that is, the sequence has only one limit.
- If
, then almost by definition we get that
. (
Is the sequence of distances).
Examples
- In
with the natural metric, The series
converges to
. And we note it as follows:
- Any space, with the discrete metric. A series
converges, only if it is eventually constant. In other words:
If and only if, We can find
that for each
,
- An example you might already know:
The space
For any p-norm induced metric, when
. Let
. and let
.
Then,
If and only if
.
Then,
Uniform Convergence
A sequence of functions
is said to be uniformly convergent on a set
if for any
, there exists an
such that when
and
are both greater than
, then
for any
.