Does a sequence of vectors whose norms converge to zero necessarily converge to the zero vector? 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?
Here is a proof that uses no theorem on normed linear spaces: Let $x_n=(a_1^{(n)},a_2^{(n)},\cdots,a_N^{(n)})$ and $r_n$ be its (usual) norm in $\mathbb R^{N}$. If $\{r_n\}$ is unbounded there is a subsequence $\{r_n'\}$ that tends to $\infty$. Now $\frac {v_n'} {r_n'} \to 0$ and we get unit vectors $(b_1^{(n')},b_2^{(n')},\cdots,b_N^{(n')})$ (obtained by normalizing $(a_1^{(n')},a_2^{(n')},\cdots,a_N^{(n')})$) such that $\sum b_i^{(n')}e_i \to 0$. But any sequence of unit vectors in $\mathbb R^{N}$ has  a subsequence which tends to  unit vector. We end up with $\sum c_i e_i=0$ where $(c_1,c_2,\cdots,c_n)$ is a unit vector contradicting linear independence of $e_i$'s. Hence $\{r_n\}$ is bounded. Now extract a convergent subsequence and take the limit in the equation $v_n=\sum a_1^{(n)}e_i$ to see that the limit of the subsquence must be $(0,0,\cdots,0)$. Now you can apply this argument to subsequences of $(a_1^{(n)},a_2^{(n)},\cdots,a_N^{(n)})$ to conclude that every subsequence of this has  a further subsequence converging to the zero vector. Hence $a_i^{(n)} \to 0$ for each $i$.

PS This approach can be used prove many of the basic theorem about finite dimensional normed linear spaces from scratch. Unfortunately it is not seen in text books. By this approach you can prove that finite dimensional subspaces are closed, that linear maps on finite dimensional spaces are continuous and all norms on finite dimensional spaces are equivalent.
Going off this question, is it possible for there to be an infinite sequence whose L2 norm goes to zero as N goes to infinity, but for every t in it's duration, the signal at t does NOT tend to zero?