Suppose I have the metric spaces $(\mathbb{R}^2,\|\cdot\|_2)$ and $(\mathbb{R}^n,\|\cdot\|_\infty)$ where $\|x-y\|_2=\sqrt{\sum_{i=1}^2 (x_i-y_i) }$ and $\|x-y\|_\infty =\max \lbrace{|x_i-y_i|}\rbrace$ for $1 \leq i \leq n$. If I have the inequality $$\|x-y\|_\infty \leq \|x-y\|_2$$ and I also know that $(\mathbb{R}^n,\|\cdot\|_2)$ is complete, can I conlcude that $(\mathbb{R}^n,\|\cdot\|_\infty)$ is also complete?
Actually my original question is to prove that $(\mathbb{R}^n,\|\cdot\|_\infty)$ is complete. If the inequality cannot imply that the metric space is complete, then can anyone guide me on how to show the completeness?
Hint: On a finite-dimensional space, all norms are equivalent. This means in your case that $c||\cdot||_{2}\leq||\cdot||_{\infty}\leq C||\cdot||_{2}$ for some absolute constants $c\leq C$. Use this to verify that a sequence $x_{n}$ converges in $(\mathbb{R},||\cdot||_{\infty})$ if and only if $x_{n}$ converges in $(\mathbb{R},||\cdot||_{2})$ which you assumed to be complete.
Note that these metrics are norms, and the claim is not true for general metrics (cf. stereographic projection, etc.).