Can we say that $X=\{x_n\}_{n\in\mathbb{N}}$ with $x_m=\sum_{i=1}^{m}(\frac{1}{i})$ is totally disconnected?

Question

Let $X=\{x_n\}_{n\in \mathbb{Z}}$ where for $m\in\mathbb{N}$, $x_m=\sum_{i=1}^{m}(\frac{1}{i})$ and $x_{-m}= \sum_{i=-m}^{0}(\frac{1}{i-1})$ given the metric inherited from $\mathbb{R}$.

Can I say that $X$ is totally disconnected?

Answer 1

Any countable subset of $\Bbb R$ is totally disconnected. So yours is too.