cauchy-preserving metric spaces

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 $$\ell_1.$$

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.