Normed vector space/Finite-dimensional/Sequence/Convergence/Basis/Exercise

Let ${}V$ be a finite-dimensional ${}{\mathbb {K} }$-vector space, and let ${}v_{n}$, ${}n\in \mathbb {N}$, denote a sequence in ${}V$. Show that the sequence converges (with respect to an arbitrary norm) if and only if for some (every) basis all component sequences converge in ${}{\mathbb {K} }$.

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