\begin{definition}[Definition:Convergent Sequence/Normed Vector Space]
Let $\struct {X, \norm {\,\cdot \,} }$ be a normed vector space.
Let $\sequence {x_n}_{n \mathop \in \N}$ be a sequence in $X$.
Let $L \in X$. 
The sequence $\sequence {x_n}_{n \mathop \in \N}$ '''converges to the limit $L \in X$''' {{iff}}:
:$\forall \epsilon \in \R_{>0}: \exists N \in \R_{>0}: \forall n \in \N: n > N \implies \norm {x_n - L} < \epsilon$
\end{definition}