What are some examples of subsets in $mathbb{R}$ that have infinitely many limit points but contain none of them? Code Answer

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My real analysis book mentions that subsets in $mathbb{R}$ may have the above properties, but doesn’t elaborate on it. I’m wondering, what are some examples and what’s the intuition? Just for context, introductory real analysis course starting on topology.

Thanks!

Answer

As a concrete exemple, take the sequence $x_{k} = frac{1}{n}$ it has a limit point but doesn’t contain it.

Then the following set has infinitely many limit points but doesn’t contain any :

$$ E = leftlbrace frac{1}{n+1}+k : nin mathbb{N}^*, kin mathbb{Z} right rbrace$$

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