# Category:Convergence

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This category contains results about Convergence.

Definitions specific to this category can be found in Definitions/Convergence.

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

Let $A \subseteq S$.

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

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

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

## Subcategories

This category has the following 13 subcategories, out of 13 total.

## Pages in category "Convergence"

The following 55 pages are in this category, out of 55 total.

### C

- Cauchy Product of Absolutely Convergent Series
- Conditions for Preservation of Covergence in Test Function Space under Differentiation
- Convergence in Indiscrete Space
- Convergence of Limsup and Liminf
- Convergence of Sequence of Test Functions in Test Function Space implies Convergence in Schwartz Space
- Convergent Sequence in Normed Division Ring is Bounded
- Convergent Sequence in P-adic Numbers has Unique Limit
- Convergent Sequence in Set of Integers
- Convergent Sequence in Set of Integers/Corollary
- Convergent Subsequence in Closed Interval
- Convergent Subsequence of Cauchy Sequence in Normed Division Ring

### D

- Definite Integral of Uniformly Convergent Series of Continuous Functions
- Derivative of Uniformly Convergent Sequence of Differentiable Functions
- Derivative of Uniformly Convergent Series of Continuously Differentiable Functions
- Dirac Comb is Distribution
- Dirichlet's Test for Uniform Convergence
- Distribution acting on Sequence of Test Functions without common Support is not Continuous

### E

- Equivalence of Definitions of Convergence in Normed Division Rings
- Equivalence of Definitions of Convergent P-adic Sequence
- Equivalence of Definitions of Convergent Sequence in Metric Space
- Existence of Radius of Convergence of Complex Power Series
- Existence of Radius of Convergence of Complex Power Series/Absolute Convergence
- Existence of Radius of Convergence of Complex Power Series/Divergence
- Expression for Set of Points at which Sequence of Functions does not Converge to Given Function

### F

### L

- Limit of Positive Real Sequence is Positive
- Limit of Sequence is Limit of Real Function
- Limit of Subsequence equals Limit of Real Sequence
- Limit of Subsequence equals Limit of Sequence
- Limit of Subsequence equals Limit of Sequence/Metric Space
- Limit of Subsequence equals Limit of Sequence/Normed Vector Space
- Limit of Subsequence equals Limit of Sequence/Real Numbers
- Logarithm of Convergent Product of Real Numbers
- Logarithm of Divergent Product of Real Numbers
- Logarithm of Infinite Product of Real Numbers

### M

### S

- Sequence is Bounded in Norm iff Bounded in Metric
- Sequence on Finite Product Space Converges to Point iff Projections Converge to Projections of Point
- Set of Points at which Sequence of Measurable Functions does not Converge to Given Measurable Function is Measurable
- Squeeze Theorem for Filter Bases