Category:Convergent Sequences
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This category contains results about Convergent Sequences.
Definitions specific to this category can be found in Definitions/Convergent Sequences.
Topological Space
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 9 subcategories, out of 9 total.
B
- Basic Null Sequences (empty)
C
- Convergent P-adic Sequences (empty)
- Convergent Sequences (Topology) (empty)
Pages in category "Convergent Sequences"
The following 8 pages are in this category, out of 8 total.
C
- Characterization of Convergence in Locally Convex Space
- Closure of Subset of Metric Space by Convergent Sequence
- Continuous Mappings preserve Convergent Sequences
- Convergence of Sequence in Discrete Space
- Convergence of Sequence in Discrete Space/Corollary
- Convergent Sequence in Particular Point Space