Is this a compact space?

Question

Let $A=\{x:d_\infty(x,0)\le 1 \}$, the subspace of the space of bounded sequences $x=(x_n)^\infty_{n=1}$, $x_n\in \mathbb{R}$, with metric $\{x:d_\infty(x,y)= sup_n |x_n-y_n| \}$. The answer says it is not compact because it is not totally bounded, but I don't see why, and I think that I can prove every sequence in this space has a convergent sub-sequence, but I am not sure what is going on. Help please.

Answer 1

For $n\in\Bbb Z^+$ let $x^{(n)}=\langle x_k^{(n)}:k\in\Bbb Z^+\rangle$ be the sequence defined by

$$x_k^{(n)}=\begin{cases} 1,&\text{if }k=n\\ 0,&\text{otherwise}\;. \end{cases}$$

Clearly $\langle x^{(n)}:n\in\Bbb Z^+\rangle$ is a sequence in $A$, and it has no convergent subsequence; indeed, any two distinct terms are distance $1$ apart.