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Problem with Proof

  1. Aug 12, 2012 #1
    Let S1 be the set that contains the natural number 1. Since S1 is finite it has no limit points.

    Let Sk be the set that contains the natural numbers less than or equal to k. Sk is finite and therefore has no limit points. The set Sk+1 contains only one more element than Sk and therefore also contains no limit points.

    Therefore the set of natural numbers contains no limit points.

    I've been told that the conclusion does not follow. Why is that the case?
  2. jcsd
  3. Aug 12, 2012 #2


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    Gold Member

    The sets [itex]S_k = \{\frac{1}{n}:1 \leq n \leq k\}[/itex] for each [itex]k \in \mathbb{N}[/itex] are finite and have no limit points, but their union does have a limit point. By your reasoning, the union should have no limit points though. Do you see the problem?
  4. Aug 12, 2012 #3
    Is it because I am trying to make a statement about an infinite set using only finite sets?
  5. Aug 12, 2012 #4
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