Definition:Convergent Sequence/Real Numbers

Definition
Let $\sequence {x_k}$ be a sequence in $\R$.

The sequence $\sequence {x_k}$ converges to the limit $l \in \R$ :


 * $\forall \epsilon \in \R_{>0}: \exists N \in \R_{>0}: n > N \implies \size {x_n - l} < \epsilon$

where $\size x$ denotes the absolute value of $x$.

Also see

 * Definition:Divergent Real Sequence
 * Definition:Unbounded Divergent Real Sequence


 * Real Number Line is Metric Space

Generalizations

 * Definition:Convergent Sequence in Real Euclidean Space
 * Definition:Convergent Sequence in Metric Space