Proof that every finite dimensional normed vector space is complete

Yes, your proof is correct. Here, I will just reword it to slightly improve clarity and precision .

Let $V$ be a vector space over $\mathbb R$ (or $\mathbb C$) with $\dim(V) = n$ and norm $\|\cdot\|$. Let $\{e_i\}_{i=1,\cdots , n}$ be a base of $V$. Suppose $v_k$ be a Cauchy sequence w.r.t. $\|\cdot\|$.

Since any two norms on a finite dimensional space are equivalent, $\|\cdot\|$ is equivalent to the $l^1$-norm $\|\cdot\|_1$. So, there are $C,D>0$ such that, for all $w\in V$, $C \|w\|_1 \leq \|w\| \leq D \|w\|_1$.

So, we have, for all $\varepsilon > 0$, there is $N$ such that, if $k,j>N$, $$ \varepsilon > \|v_j - v_k\| \geq C \|v_j - v_k\|_1 = C \sum_{i=1}^n |v_{ji} - v_{ki}| \geq C |v_{ji} - v_{ki}|$$ for each $1 \leq i \leq n$. Hence $v_{ki}$ is a Cauchy sequence in $\mathbb R$ (or $\mathbb C$) for each $i$. Since $\mathbb R$ (or $\mathbb C$) is complete, there is $u_i$ in $\mathbb R$ (or $\mathbb C$) such that $u_i = \lim_{k \to \infty} v_{ki} $, for each $i$. Let $u = (u_1, \dots , u_n) = \sum_i u_i e_i$. Then, it is clear that, $u$ is in $V$.

Let us prove $\lim_{k \to +\infty} \|v_k - u\| = 0$:

$$ \lim_{k \to +\infty} \|v_k - u\| \leq D \lim_{k \to +\infty} \|v_k - u\|_1 = D \lim_{k \to +\infty} \sum_{i=1}^n |v_{ki} - u_i| = D \sum_{i=1}^n \lim_{k \to +\infty} |v_{ki} - u_i|=0$$

I think there is a simple way to demonstrate this.

Let $ \left(E, \| \| \right)$ be a $\mathbb{K}$ finite dimensionial vector space. Consider the following application:

\begin{aligned} \mathbb { K }^ { n } & \rightarrow E \\ \left( \lambda _ { 1 } , \ldots , \lambda _ { n } \right) & \mapsto \sum _ { i = 1 } ^ { n } \lambda _ { i } e _ { i } \end{aligned} which is a linear isometric bijection (thus, a homeomorphism) between $ \left(\mathbb{ K }^ { n }, \| \| _\infty\right)$ and $ \left(E, \| \|_{\infty} \right)$.

The pre-image of a complete space by a uniformly continuous and bijective function being complete and $ \left(\mathbb{ K }^ { n }, \| \| _\infty\right)$ being a complete space, then $ \left( E, \| \| _\infty\right)$ is complete. Since all norms on a finite dimensional space are equivalent, it folows that $ \left(E, \| \| \right)$ is complete.