# Determining whether a subspace of the metric space of real sequences is separable

Let $$X=\left\{(a_n)_{n \in \mathbb N} \in \mathbb R^N : \exists n_0 \in \mathbb N\, \forall n\ge n_0 \big(a_n\le \sqrt{n}\big)\right\}$$ with the metric $$d\big((a_n)_{n \in \mathbb N},(b_n)_{n \in \mathbb N}\big)=\sup_{n \in \mathbb N}\frac{|a_n-b_n|}{n}$$

1. Determine if $(X,d)$ is separable

2. Prove that for every Cauchy sequence in $X$ there exists a sequence of real terms such that the Cauchy sequence converges to that one but that $X$ is not complete.

Can anyone give me a hint?

With this definition of $X$, the function $d$ isn’t defined on all of $X\times X$: for example, if $z$ is the zero sequence and $x=\langle -n^2:n\in\Bbb N\rangle$, then $z,x\in X$, but $$\sup_{n\in\Bbb N}\frac{|0-(-n^2)|}n=\sup_{n\in\Bbb N}n$$ doesn’t exist. I’m going to assume that the last condition in the definition of $X$ was supposed to be that $|a_n|\le\sqrt{n}$. Then if $x=\langle x_n:n\in\Bbb N\rangle,y=\langle y_n:n\in\Bbb N\rangle\in X$, then there is an $m\in\Bbb N$ such that $$\frac{|x_n-y_n|}n\le\frac{2\sqrt{n}}n=\frac2{\sqrt{n}}$$ for all $n\ge m$, and $d(x,y)$ is defined.
1. Consider rational sequences that are eventually $0$.
2. For $n,k\in\Bbb N$ let $$x_k^{(n)}=\begin{cases}2\sqrt{k},&\text{if }k\le n\\0,&\text{if }k>n\;,\end{cases}$$ and let $x^{(n)}=\left\langle x_k^{(n)}:k\in\Bbb N\right\rangle$. Show that $\left\langle x^{(n)}:n\in\Bbb N\right\rangle$ is a Cauchy sequence in $X$ that has no limit in $X$. There is a sequence $x\in\Bbb R^{\Bbb N}\setminus X$ such that $d\big(x^{(n)},x\big)$ is defined for each $n\in\Bbb N$ and converges to $0$ as $n\to\infty$; what is it?
More generally, show that if $\left\langle x^{(n)}:n\in\Bbb N\right\rangle$ is any Cauchy sequence in $X$, where $x^{(n)}=\left\langle x_k^{(n)}:k\in\Bbb N\right\rangle$ for each $n\in\Bbb N$, then $\left\langle x_k^{(n)}:n\in\Bbb N\right\rangle$ is a Cauchy sequence in $\Bbb R$ for each $k\in\Bbb N$. Use this observation to show that there is an $x\in\Bbb R^{\Bbb N}$ such that $d\big(x^{(n)},x\big)$ is defined for each $n\in\Bbb N$ and converges to $0$ as $n\to\infty$.