Definition:Convergent Sequence/Metric Space/Definition 1

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:

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

Also see

• Equivalence of Definitions of Convergent Sequence in Metric Space

Sources

• 1953: Walter Rudin: Principles of Mathematical Analysis ... (previous) ... (next): $3.1$