Let $\mathbb{F}\in\{\mathbb{R},\mathbb{C}\}$, and let $(V,\|\cdot\|_V)$ be a finite-dimensional [normed vector space](/page/Normed%20Vector%20Space) over $\mathbb{F}$. Then $(V,\|\cdot\|_V)$ is [complete](/page/Complete%20Metric%20Space): every [Cauchy sequence](/page/Cauchy%20Sequence) $(x_k)_{k=1}^{\infty}$ in $V$ with respect to $\|\cdot\|_V$ converges in $\|\cdot\|_V$ to some element $x\in V$. Equivalently, $(V,\|\cdot\|_V)$ is a [Banach space](/page/Banach%20Space).