Let $V$ be an $N$-dimensional normed vector space, and let $(v\_n)$ be a sequence of vectors such that $\lim (\left \| v\_n \right \|)=0$. Given some basis $\{e^1,e^2,..., e^N \}$, we can write each $v\_n$ as $a\_1^{(n)}e^1+a\_2^{(n)}e^2 +\dots + a\_N^{(n)}e^N$. 

Is it necessarily true that for each fixed $k \in \{1, 2, \dots, N\}$, $\lim\_{n \to \infty} a\_k^{(n)}=0$ ? If so, how can we prove this using only the axioms for a generic norm?