Definition:Convergent Sequence/Normed Division Ring/Definition 3

Definition

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:

the real sequence $\sequence {\norm {x_n - x} }$ converges to $0$ in the reals $\R$

Also see

• Definition:Convergent Real Sequence

Sources

• 2007: Svetlana Katok: p-adic Analysis Compared with Real ... (previous) ... (next): $\S 1.2$: Normed Fields: Definition $1.7$