Definition:Convergent Sequence/Metric Space/Definition 4
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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
Sources
- 1953: Walter Rudin: Principles of Mathematical Analysis ... (previous) ... (next): $\S 3$ Numerical Sequences and Series, Theorem $3.2a$