And also I need to write a set unbounded that has only one point of accumulation and all others isolated point. I know that the set of numbers naturals has infinite points but don't have any point of accumulation, but I don't know which is those sets of the question. I have like the definition of point accumulation :

For the set X ⊆ R, the neighborhood of a Vε (a) ∩X-{a} ≠ ∅, ε > 0 is the variation of neighborhood

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    $\begingroup$ You want a set of real numbers with exactly one accumulation point? Can you think of a decreasing set of real numbers that converges to $0$, say? $\endgroup$ – lulu Mar 30 '18 at 14:17
  • $\begingroup$ Do you have the definition of "accumulation point"? $\endgroup$ – Eric Towers Mar 30 '18 at 14:25
  • $\begingroup$ So, I can write something like this: 0 < = x < 1 ?? $\endgroup$ – Juliana Rodrigues Mar 30 '18 at 14:26
  • $\begingroup$ For the set X ⊆ R, the neighborhood of a Vε (a) ∩X-{a} ≠ ∅ **ε > 0, is the variation of neighborhood ** $\endgroup$ – Juliana Rodrigues Mar 30 '18 at 14:38
  • $\begingroup$ The English term is "isolated point", not "insulated point". Because somebody many years ago said so. $\endgroup$ – DanielWainfleet Mar 30 '18 at 17:33

Any convergent sequence form a set with only a point of accumulation so you can take for exemple $ A = \{ \frac{1}{n} ; n \in \mathbb{N} \} $

and an unbounded set with the same property is $ A \cup \mathbb{N} $

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    $\begingroup$ Thanks a lot!!!! $\endgroup$ – Juliana Rodrigues Mar 30 '18 at 17:17

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