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Proving the set of all cluster points is closed

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if $A$ is a subset of $\mathbb{R}^n$. Can I prove that "the set of all cluster points of A is closed" using the following:

  • A closed subset is a subset that contains all its limit points.
  • Every cluster point is a limit point (Not vice versa though).
  • A has all cluster points means that it includes all its limits point and henceA is closed.

Is this an enough proof ?


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