Cauchy sequences the space of binary sequences with the metric $\sum 2^{-k}|x_k-y_k| $

Question

Let's consider the following metric space $(X,d)$, where:

$X = \{ \ x = (x^1,x^2,x^3,\ldots,x^k,\ldots)\ \mid \ x^j \in \{0,1\}\ \forall j \geq 1\ \}$

$d(x,y) = \sum\limits_{k=1}^{\infty} \frac{1}{2^{k}} | x^{k} - y^{k} |$

This is the space of all sequences consisting of $1$'s and $0$'s. This is a metric space, it's easy to see why.

I'm examining the Cauchy sequences in this metric space, and it seems to me that the Cauchy sequences in this space are constant. Meaning, I think that the Cauchy sequences $\{x_{n}\}_{n=1}^{\infty}$ in this metric space, are those sequences which satisfy $x_{n}^k = x_{m}^k$ for all $n,m \geq N$, for some $N \in \mathbb{N}$.

Is this correct?

Answer 1

KoliG gave an example of a nonconstant Cauchy sequence: let $x_n$ be the sequence with $n$th entry $1$ and other entries $0$. Since $d(x_n,x_m)=2^{-n}+2^{-m}$ for $n\ne m$, this is a Cauchy sequence. It converges to the zero sequence.