No Limit Points in Natural Numbers: Problem with Proof?

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Discussion Overview

The discussion centers around the concept of limit points in the context of natural numbers and finite sets. Participants explore the implications of using finite sets to draw conclusions about infinite sets, particularly focusing on the set of natural numbers.

Discussion Character

  • Debate/contested
  • Conceptual clarification

Main Points Raised

  • One participant asserts that the set of natural numbers contains no limit points based on the properties of finite sets.
  • Another participant challenges this conclusion by presenting a counterexample involving finite sets whose union has a limit point, questioning the validity of the original reasoning.
  • A third participant suggests that the issue may stem from attempting to generalize properties of finite sets to an infinite set.
  • A reference is made to an external thread for further clarification on the topic.

Areas of Agreement / Disagreement

Participants do not appear to reach a consensus, as there are competing views regarding the application of finite set properties to infinite sets.

Contextual Notes

The discussion highlights the limitations of reasoning about infinite sets based solely on finite examples, but does not resolve the underlying mathematical questions or assumptions involved.

Dschumanji
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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?
 
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Dschumanji said:
I've been told that the conclusion does not follow. Why is that the case?

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?
 
jgens said:
Do you see the problem?
Is it because I am trying to make a statement about an infinite set using only finite sets?
 

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