# Definition:Convergent Sequence/Normed Vector Space

## Definition

Let $\tuple {X, \norm {\,\cdot \,}}$ 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 $L \in X$ if and only if:

$\forall \epsilon \in \R_{>0}: \exists N \in \N: \forall n \in \N: n > N \implies \norm {x_n - L} < \epsilon$