Show metrics induced by ${\parallel \cdot \parallel}_1 ,{\parallel \cdot \parallel}_2, {\parallel \cdot \parallel}_{\infty}$ are equivalent

Question

Show that the metrics induced by ${\parallel \cdot \parallel}_1$ ,${\parallel \cdot \parallel}_2$ and ${\parallel \cdot \parallel}_{\infty}$ on ${\mathbb{R}}^n$ are equivalent.

Two metrics $d$ and $\rho$ on a set M are equivalent if they generate the same convergent sequences; that is, $d(x_n,x) \to 0$ iff $\rho(x_n,x)\to 0$.

I have proved that ${\parallel \cdot \parallel}_{\infty}\le{\parallel \cdot \parallel}_2\le{\parallel \cdot \parallel}_1$. So the one direction is easy. If ${\parallel x_n-x \parallel}_1\to 0$, then ${\parallel x_n-x \parallel}_2 \to 0$ and ${\parallel x_n-x \parallel}_{\infty}\to0$.

How about the other direction? If ${\parallel x_n-x \parallel}_{\infty}\to 0$, how can I get ${\parallel x_n-x \parallel}_2$ and ${\parallel x_n-x \parallel}_1 \to 0$?

Answer 1

Hint: Observe \begin{align} \|x\|_\infty \leq \| x \|_2 \leq \|x\|_1 \leq n\|x\|_\infty. \end{align} where $n$ is the dimension of your euclidean space.