Definition:Convergent Sequence/Metric Space/Definition 4

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:

for every $\epsilon \in \R{>0}$, the open $\epsilon$-ball about $l$ contains all but finitely many of the $x_k$.

Also see

• Equivalence of Definitions of Convergent Sequence in Metric Space

Sources

• 1953: Walter Rudin: Principles of Mathematical Analysis ... (previous) ... (next): $\S 3$ Numerical Sequences and Series, Theorem $3.2a$