Definition:Convergent Sequence/Metric Space/Definition 2

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:

$\forall \epsilon > 0: \exists N \in \R_{>0}: \forall n \in \N: n > N \implies x_n \in \map {B_\epsilon} l$

where $\map {B_\epsilon} l$ is the open $\epsilon$-ball of $l$.

Also see

• Equivalence of Definitions of Convergent Sequence in Metric Space

Sources

• 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $4$: The Hausdorff condition: $4.1$: Motivation: Definition $4.1.1$