Convergence in norm

Convergence of a sequence $(x_n)$ in a normed vector space $X$ to an element $x$, defined in the following way: $x_n \rightarrow x$ if $$\text{\left\| x_n - x \right\| \rightarrow 0 as n\rightarrow\infty.}$$ Here $\left\|\cdot\right\|$ is the norm in $X$.