Category:Convergent Sequences in Normed Division Rings
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This category contains results about Convergent Sequences in Normed Division Rings.
Definitions specific to this category can be found in Definitions/Convergent Sequences in Normed Division Rings.
Let $\struct {R, \norm {\, \cdot \,} }$ be a normed division ring.
Let $\sequence {x_n} $ be a sequence in $R$.
The sequence $\sequence {x_n}$ converges to the limit $x \in R$ in the norm $\norm {\, \cdot \,}$ if and only if:
- $\forall \epsilon \in \R_{>0}: \exists N \in \R_{>0}: \forall n \in \N: n > N \implies \norm {x_n - x} < \epsilon$
Subcategories
This category has the following 3 subcategories, out of 3 total.
Pages in category "Convergent Sequences in Normed Division Rings"
The following 7 pages are in this category, out of 7 total.
C
- Combination Theorem for Sequences in Normed Division Rings
- Convergent Sequence in Normed Division Ring is Bounded
- Convergent Sequence in Normed Division Ring is Cauchy Sequence
- Convergent Sequence is Cauchy Sequence/Normed Division Ring
- Convergent Sequence with Finite Elements Prepended is Convergent Sequence
- Convergent Subsequence of Cauchy Sequence in Normed Division Ring
- Convergent Subsequence of Cauchy Sequence/Normed Division Ring