Example of a countable compact set in the real numbers Can someone give me an example or a hint to come up with a countable compact set in the real line with infinitely many accumulation points?
Thank you in advance!

What about if we define $H = \{ \frac{1}{n} : n\in \mathbb{N}\} \cup \{0\}$, a sort of standard countable compact set with 0 at its sole limit point, then define your countable compact set to be:
$$S = \{ x + y \mid x, y \in H\}.$$
To unpack the thought behind this definition:


*

*This set $S$ is countable.

*This set $S$ is bounded, because its smallest member is 0 and its largest member is 1+1 = 2.

*Each $x \in H$ is a limit point of $S$. If we fix $x$ and vary $y\in H$ in the definition of $S$, we can see that $x$ is a limit point of $S$.

*This set $S$ is closed (and therefore compact, because it is a bounded subset of the real line.)  ($S$ is closed because it is the sum of two compact subsets of $\mathbb{R}$ and is therefore closed in $\mathbb{R}$.)

