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Let $d, p$ be two metrics on a nonempty set $X$. Say $d\sim p$ if for any $(x_n)\subseteq X$ and $x\in X, \lim_{n} x_n = x$ in $(X,d)$ iff $\lim_n x_n = x$ in $(X,p)$. Say $d\approx p$ if $\exists m, M > 0$ so that $m d(x,y) \leq p(x,y) \leq Md(x,y)$ for $x,y \in X.$

Give an example of an $X$ and two metrics $d,p$ on $X$ with $d\not \approx p$ but such that a sequence $(x_n)_{n=1}^\infty \subseteq X$ is Cauchy in $(X,d)$ iff it is Cauchy in $(X,p).$

Give an example of an $X$ and two metrics $d,p $ on $X$ with $d\sim p$ and so that there is a sequence $(x_n)_{n=1}^\infty \subseteq X$ that is Cauchy in one of $(X,d)$ or $(X,p)$ but not the other.

I know that on $\ell_1, \lVert \cdot \rVert_r \not \approx \lVert \cdot \rVert_p,$ where $1\leq p < r \leq \infty$ and $\lVert x\rVert_r := (\sum_{i\in\mathbb{N}} |x_i|^r)^{1/r}$ for $x= (x_i)_{i\in\mathbb{N}}\in \ell_1,$ but it seems that there are Cauchy sequences in $(\ell_1, \lVert \cdot \rVert_r)$ that aren't Cauchy in $(\ell_1, \lVert \cdot \rVert_p).$ For example, take the sequence $(x_n)_{n\in\mathbb{N}}, x_n = (1, \frac{1}{2^{1/p}}, \frac{1}{3^{1/p}},\cdots, \frac{1}{n^{1/p}}, 0,0, \cdots)$ for each $n.$ It's not Cauchy in $(X, p)$ because if one fixes $n\in\mathbb{N}$, then as $m\to \infty, \lVert x_n - x_m\rVert_p =(\sum_{i=n+1}^\infty \frac{1}i)^{1/p},$ which diverges. It is Cauchy in $(X,r)$ because for any $\epsilon > 0,$ there exists $N\in\mathbb{N}$ so that for $n\geq N, (\sum_{i=n+1}^\infty \frac{1}{i^{r/p}})^{1/r} < \epsilon$ (as $ \sum_{i=1}^\infty \frac{1}{i^{r/p}})^{1/r} < \infty$ by the $p$-series test as $r/p > 1$). However, this may not be useful for the second part as I'm not sure if $\lVert \cdot \rVert_p\sim \lVert \cdot \rVert_r$ in $\ell_1.$

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Let $X=\Bbb Z^+$. Let $d$ be the discrete metric on $X$:

$$d(m,n)=\begin{cases} 1,&\text{if }m\ne n\\ 0,&\text{if }m=n\,. \end{cases}$$

Let $p$ be the usual Euclidean metric: $p(m,n)=|m-n|$. These metrics both generate the discrete topology, so the Cauchy sequences in both are precisely the sequence that are eventually constant, but clearly $d\not\approx p$.

For the second problem let $X=\Bbb R$, let $d(x,y)=|\tan^{-1}x-\tan^{-1}y|$ for all $x,y\in\Bbb R$, and let $p$ be the usual Euclidean metric. Let $\langle x_n:n\in\Bbb N\rangle$ be a sequence in $\Bbb R$. Then

$$\lim_{n\to\infty}|x_n-x|=0$$

iff

$$\lim_{n\to\infty}|\tan^{-1}x_n-\tan^{-1}x|=0\,,$$

so $d\sim p$, but the sequence $\langle n:n\in\Bbb N\rangle$ is $d$-Cauchy and not $p$-Cauchy.

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