Question about $l_\infty$ as a complete metric space

Question

I have question about the answer by Brian M. Scott to this question Is the set of all bounded sequences complete?

In his answer, $\langle x^n : n \in \mathbb{N} \rangle$ is a sequence object. $x_n$ is a single real number. Therefore, I don't understand the concept of writing $x^n = \langle x_k^n : k \in \mathbb{N} \rangle$ as another sequence object.

I know that a real number is an equivalence class of Cauchy sequence of rational numbers but I don't think that is what is happening here since he defines $x^n = \langle x_k^n : k \in \mathbb{N} \rangle$ as a sequence in reals.

I know he is right; I confirmed this from other places. I just cannot understand the underlying concept.

Answer 1

What Brian writes is this:

• he is working with a sequence $\langle x^n:n\in\mathbb N\rangle$ of elements of $\ell^\infty$; so, each $x^n$ is an element of $\ell^\infty$;
• if you fix a $n\in\mathbb N$, then, since $x^n\in\ell^\infty$, $x^n$ is a sequence$$\left\langle x_k^n:k\in\mathbb N\right\rangle$$of real numbers.

The fact “that a real number is an equivalence class of Cauchy sequence of rational numbers” is not used at all in Brian's answer.