\begin{definition}[Definition:Convergent Sequence/Normed Division Ring/Definition 1]
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 $\norm {\, \cdot \,}$''' {{iff}}:
:$\forall \epsilon \in \R_{>0}: \exists N \in \R_{>0}: \forall n \in \N: n > N \implies \norm {x_n - x} < \epsilon$
Category:Definitions/Convergent Sequences in Normed Division Rings
\end{definition}