Definition:Convergence

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[edit] Convergent Sequence

[edit] Topological Space

Let T = \left({A, \vartheta}\right) be a topological space.

Let \left \langle {x_k} \right \rangle be a sequence in T.


Then \left \langle {x_k} \right \rangle converges to the limit \alpha \in T if:

for any open set U \subseteq T such that \alpha \in U: \exists N \in \R: n > N \implies x_n \in U.


This can be alternatively stated:

\left \langle {x_k} \right \rangle converges to the limit l \in T if:

every open set in X containing \alpha contains all but a finite number of terms of \left \langle {x_n} \right \rangle.


Such a sequence is convergent.


[edit] Metric Space

Let \left({X, d}\right) be a metric space.

Let \left \langle {x_k} \right \rangle be a sequence in X.


Then \left \langle {x_k} \right \rangle converges to the limit l iff:

\forall \epsilon > 0: \exists N \in \R: n > N \implies d \left({x_n, l}\right) < \epsilon

Or equivalently, using the definition of neighborhood:

\forall \epsilon > 0: \exists N \in \R: n > N \implies x_n \in N_\epsilon \left({l}\right)


We can write "x_n \to l as n \to \infty", or \lim_{n \to \infty} x_n \to l.

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 \Q, \R, \C, and the metric d is the usual (Euclidean) metric, the condition on convergence becomes:

\left \langle {x_k} \right \rangle converges to the limit l iff:

\forall \epsilon > 0: \exists N \in \R: n > N \implies \left|{x_n - l}\right| < \epsilon

where \left|{x}\right| is the modulus of x.

The validity of this definition derives from the fact that:


[edit] Convergent Series

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

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

Let \left \langle {s_N} \right \rangle be the sequence of partial sums of \sum_{n=1}^\infty a_n.

It follows that \left \langle {s_N} \right \rangle 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.


[edit] Convergent Function

[edit] Convergence of a Function on a 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.


[edit] Convergence of Real and Complex Functions

As:

the definition holds for real and complex functions.


[edit] Convergent Filter

Let \left({X, \vartheta}\right) be a topological space.

Let \mathcal F be a filter on X.


Then \mathcal F converges to a point x \in X if:

\forall N_x \subseteq X: N_x \in \mathcal F

where N_x is a neighborhood of x.


That is, a filter is convergent to a point x if every neighborhood of x is an element of that filter.

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