# Definition:Limit of Sequence/Normed Vector Space

## Definition

Let $M = \struct {X, \norm {\, \cdot \,}}$ be a normed vector space.

Let $L \in X$.

Let $\sequence {x_n}_{n \mathop \in \N}$ be a sequence in $X$.

Let $\sequence {x_n}_{n \mathop \in \N}$ converge to $L$.

Then $L$ is a limit of $\sequence {x_n}_{n \mathop \in \N}$ as $n$ tends to infinity which is usually written:

$\displaystyle L = \lim_{n \mathop \to \infty} x_n$