Definition:Convergent Sequence/Note on Domain of N

Definition
Let $\left \langle {x_k} \right \rangle$ be a sequence.
 * $\displaystyle \lim_{n \to \infty} x_n \to l$

be the limit of $\left \langle {x_k} \right \rangle$.

That is:
 * $\forall \epsilon \in \R_{>0}: \exists N \in \R_{>0}: n > N \implies d \left({x_n, l}\right) < \epsilon$

Note that some sources insist that $N \in \N$ but this is not strictly necessary and can make proofs more cumbersome.