How to show $\text{Given any sequence} (x^{(n)}) _{n\in\mathbb{N}}\text{ converges to } x \text{ in } (X ,|| •||), \text {X :finite dimensional NLS?}$

Let $\lim_{n\to\infty}\|x(n)-x\|=0$ where $x(n)=\sum_{j=1}^de_jx_{n,j}$ and $x=\sum_{j=1}^de_jx_j.$ For simplicity of notation, let $z(n)=x(n)-x=\sum_{j=1}^de_j(x_{n,j}-x_j)=\sum_{j=1}^de_jz_{n,j}.$ Then $\lim_{n\to\infty}\|z(n)\|=0$ and we wish to show that $\lim_{n\to\infty}z_{n,j}=0$ for each $j\le d.$

Suppose by contradiction that $z(n)$ does not converge co-ordinate-wise to $0.$ Take $r>0$ such that tne set $A=\{n\in\Bbb N:\max_{j\le d}|z_{n,j}|>r\}$ is infinite. For each $n\in A$ take some $j(n)\le d$ such that $|x_{n,j(n)}|=\max_{j\le d} |x_{n,j}|.$ Now $d$ is finite so we may take some fixed $j'\le d$ such that the set $B=\{n\in A: j(n)=j'\}$ is infinite.

Consider the sequence $S_B=(\,(|z_{n,j'}|^{-1}\cdot z_{n,1},...,|z_{n,j'}|^{-1}\cdot z_{n,d})\,)_{n\in B}=((w_{n,1},...,w_{n,d}))_{n\in B}$ of members of $\Bbb R^d$ or $\Bbb C^d.$

We have $\forall n\in B\,\forall j\le d\,(|w_{n,j}|\le 1).$ Therefore there exists an infinite $C\subseteq B$ such that $\lim_{n\to\infty ;n\in C}w_{n,j}=w_j$ exists for each $j\le d$. This is a finite-dimensional generalization of the fact that a bounded sequence in $\Bbb R$ or $\Bbb C$ has a convergent subsequence.

We have $|z_{n,j'}|>r$ for $n\in C$ so we have $$(\bullet)\quad 0=r^{-1}\lim_{n\to\infty ;n\in C}\|z_n\|=$$ $$=\lim_{n\to\infty ;n\in C}r^{-1}\|z_n|\ge$$ $$\ge \lim_{n\to\infty ;n\in C}|w_{n,j'}|^{-1}\|z_n\|=$$ $$=\lim_{n\to\infty ;n\in C}\|\sum_{j=1}^de_jw_{n,j}\|.$$ And the sequence $T=(\sum_{j=1}^de_jw_{n,j})_{n\in C}$ converges co-ordinate-wise to $w=\sum_{j=1}^de_jw_j$. So by the first part (proved in your Q), $T$ converges in norm to $w$. And by $(\bullet)$ we must have $w=0.$

BUT $w_{n,j'}=|z_{n,j'}|^{-1}z_{n,j'}$ so we have $|w_{j'}|=\lim_{m\to\infty ;n\in C}|w_{n,j'}|=1.$ Therefore $$0=w=\sum_{j=1}^dw_je_j$$ but at least one co-ordinate of $w$ (namely $w_{j'}$) is non-zero, and this contradicts the linear independence of $\{e_1,...,e_d\}.$

Remark: The linear independence of $\{e_1,...,e_d\}$ must be used eventually. Here it is used only at the finish.