Definition:Convergence

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Contents 1 Sequence 1.1 Topological Space 1.2 Metric Space 1.2.1 Standard Number Fields 1.3 Divergent Sequence 1.4 Comment 1.5 Note 2 Series 2.1 Divergent Series 3 Functions 3.1 Convergence of a Function on a Metric Space 3.2 Convergence of Real and Complex Functions 3.3 Divergent Function 4 Filters

[edit] Sequence

[edit] Topological Space

Let be a topological space.

Let be a sequence in T.

Then converges to the limit if, for any open set such that , .

Such a sequence is convergent.

[edit] Metric Space

Let be a metric space.

Let be a sequence in X.

Then converges to the limit l iff:

Or equivalently, using the definition of neighborhood:

We can write " as ", or .

This is voiced "As n tends to infinity, x_n tends to (the limit) l."

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 X is one of the standard number fields , and the metric d is the usual (Euclidean) metric, the condition on convergence becomes:

converges to the limit l iff:

where is the modulus of x.

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] Divergent Sequence

A sequence which is not convergent is divergent.

[edit] Comment

The sequence can be thought of as a set of approximations to the number l, in which the higher the n the better the approximation.

The distance between x_n and l can then be thought of as the error arising from approximating l by x_n.

Note the way the definition is constructed.

"Given any value of ε, however small, we can always find a value of N such that ..."

If you pick a smaller value of ε, then (in general) you would have to pick a larger value of N - but the implication is that, if the sequence is convergent, you will always be able to do this.

Note also that N depends on ε. That is, for each value of ε we (probably) need to use a different value of N.

[edit] Note

Some sources insist that but this is not strictly necessary and can make proofs more cumbersome.

[edit] Series

Let S be one of the standard number fields .

Let be a series in S.

Let be the sequence of partial sums of .

It follows that can be treated as a sequence in the metric space S.

If as , the series converges to the sum s.

[edit] Divergent Series

A series which is not convergent is divergent.

[edit] Functions

[edit] Convergence of a Function on a Metric Space

Let and be metric spaces.

Let c be a limit point of M₁.

Let be a mapping from A₁ to A₂ defined everywhere on A₁ except possibly at c.

Let tend to the limit L as x tends to c.

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

[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] Divergent Function

There are multiple ways that a function can be divergent. Here are some samples:

• Let be such that:

provided c < x < c + δ

Then (using the language of limits), as .

• Let be such that:

Then x converges to neither 0 nor 1 and hence is divergent (although, it needs to be noted, not to infinity).

[edit] Filters

A filter on a topological space X is said to converge to a point if for every neighborhood U of x.