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This seems to be a bit of an odd one. I have worked out a possible answer, but I have a feeling I am going about this the wrong way. Help would be appreciated.

Find $m,M\in \mathbb{R}$ so that for every $x\in R^2$,

$$m||x||_2 \leq ||x||_\infty \leq M||x||_2$$

I tried plugging in the $||x||_2$ norm and solving for $m$, which got me

$m\leq max{|x_1|,|x_2|}(x_1^2+x_2^2)^{-1/2}$

This just seems off to me...

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Hint to find $M$: note that if $x = (a,b)$, then $\|x\|_2 = (a^2 + b^2)^{1/2}$ and $\|x\|_{\infty} = \max(|a|, |b|)$. Note that $a^2$ and $b^2$ are both $\leq a^2 + b^2$.

Hint to find $m$: note that $a^2 + b^2$ is no larger than $2\cdot\max(a^2, b^2)$.

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