Category:Definitions/Convergence
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This category contains definitions related to Convergence.
Related results can be found in Category: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 8 subcategories, out of 8 total.
A
- Definitions/Absolute Convergence (18 P)
C
- Definitions/Cauchy Criterion (4 P)
- Definitions/Cauchy Sequences (5 P)
- Definitions/Convergent Series (10 P)
U
Pages in category "Definitions/Convergence"
The following 72 pages are in this category, out of 72 total.
C
- Definition:Compact Convergence
- Definition:Convergence
- Definition:Convergence in Mean
- Definition:Convergence of Product
- Definition:Convergent Complex Function
- Definition:Convergent Complex Sequence
- Definition:Convergent Continued Fraction
- Definition:Convergent Mapping
- Definition:Convergent Mapping in Metric Space
- Definition:Convergent Mapping/Complex Function
- Definition:Convergent Mapping/Metric Space
- Definition:Convergent Mapping/Real Function
- Definition:Convergent P-adic Sequence
- Definition:Convergent Product
- Definition:Convergent Product/Arbitrary Field
- Definition:Convergent Rational Sequence
- Definition:Convergent Real Sequence
- Definition:Convergent Sequence
- Definition:Convergent Sequence in Normed Vector Space
- Definition:Convergent Sequence in Strong Operator Topology
- Definition:Convergent Sequence in Uniform Operator Topology
- Definition:Convergent Sequence in Weak Operator Topology
- Definition:Convergent Sequence/Analysis
- Definition:Convergent Sequence/Complex Numbers
- Definition:Convergent Sequence/Metric Space
- Definition:Convergent Sequence/Normed Division Ring
- Definition:Convergent Sequence/Normed Vector Space
- Definition:Convergent Sequence/Note on Domain of N
- Definition:Convergent Sequence/P-adic Numbers
- Definition:Convergent Sequence/P-adic Numbers/Definition 1
- Definition:Convergent Sequence/P-adic Numbers/Definition 2
- Definition:Convergent Sequence/P-adic Numbers/Definition 3
- Definition:Convergent Sequence/P-adic Numbers/Definition 4
- Definition:Convergent Sequence/Rational Numbers
- Definition:Convergent Sequence/Real Numbers
- Definition:Convergent Sequence/Test Function Space
- Definition:Convergent Sequence/Topology
- Definition:Convergent Sequence/Topology/Definition 1
- Definition:Convergent Sequence/Topology/Definition 2
- Definition:Convergent Series
- Definition:Convergent Series/Number Field