Definition:Convergent Sequence/Metric Space/Definition 3

Definition

Let $M = \struct {A, d}$ be a metric space or a pseudometric space.

Let $\sequence {x_k}$ be a sequence in $A$.

$\sequence {x_k}$ converges to the limit $l \in A$ if and only if:

$\ds \lim_{n \mathop \to \infty} \map d {x_n, l} = 0$

Also see

• Equivalence of Definitions of Convergent Sequence in Metric Space

Sources

• 1975: Bert Mendelson: Introduction to Topology (3rd ed.) ... (previous) ... (next): Chapter $2$: Metric Spaces: $\S 5$: Limits: Definition $5.2$