Category:Definitions/Convergent Sequences in Normed Division Rings

This category contains definitions related to Convergent Sequences in Normed Division Rings.
Related results can be found in Category: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:

$\forall \epsilon \in \R_{>0}: \exists N \in \R_{>0}: \forall n \in \N: n > N \implies \norm {x_n - x} < \epsilon$