\begin{definition}[Definition:Convergent Sequence/Note on Domain of N]
Let $\sequence {x_k}$ be a sequence.
:$\ds \lim_{n \mathop \to \infty} x_n \to l$
be the limit of $\sequence {x_k}$.
That is:
:$\forall \epsilon \in \R_{>0}: \exists N \in \R_{>0}: n > N \implies \map d {x_n, l} < \epsilon$
Some sources insist that $N \in \N$ but this is not strictly necessary and can make proofs more cumbersome.
\end{definition}