Definition:Convergent Sequence/Real Numbers

Definition
Let $\left \langle {x_k} \right \rangle$ be a sequence in $\R$.

The sequence $\left \langle {x_k} \right \rangle$ converges to the limit $l$ iff:


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

where $\left\vert{x}\right\vert$ denotes the absolute value of $x$.

Also see

 * Definition:Divergent Real Sequence
 * Definition:Divergent Real Sequence to Infinity
 * Real Number Line is Metric Space

Generalizations

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