Prove $\mathbb R^n$ is complete with respect to non-standard norms Here is a proof (page 6) for the completeness of $\mathbb R^n$ with respect to the standard Euclidean norm. 
I do understand the proof but I want to try to do the proof by myself for the completeness of $\mathbb R^n$ with respect to the following norm:
$||(\lambda_1, \lambda_2, \dots , \lambda_n)||_∞ := \max(|\lambda_1|, |\lambda_2|, ...|\lambda_n|)$
May I please ask someone to briefly tell me how to prove that with respect to this particular norm? (by using the definition of completeness, namely, every Cauchy sequence converges) Thanks so much. 
I do understand that all norms are equivalent But I just want to know how to prove that with respect to this particular norm directly using the method of showing every Cauchy sequence converges.

Suppose that $(x^k)_{k \ge 0}$ is a Cauchy sequence in $(\mathbb{R}^n, \|-\|_\infty)$. Then for all $\epsilon >0$ there is some $N$ such that for all $k, l \ge N$, $\|x^k - x^l\|_\infty < \epsilon$. Then by definition of $\|-\|_\infty$, for all coordinates $1 \le i \le n$, $|x^k_i - x^l_i| < \epsilon$ for $k,l \ge N$. It follows that $\{x^k_i\}_{k \ge 0}$ is a Cauchy sequence in $\mathbb{R}$, and hence converges to some $x_i$. It follows easily that $\lim_{k \to \infty} x^k = (x_1, \dots, x_n)$.
