No Limit Points in Natural Numbers: Problem with Proof?

[Dschumanji]

Let S₁ be the set that contains the natural number 1. Since S₁ is finite it has no limit points.

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

 

[jgens]

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

The sets S_k = \{\frac{1}{n}:1 \leq n \leq k\} for each k \in \mathbb{N} 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?

 

[Dschumanji]

Do you see the problem?

Is it because I am trying to make a statement about an infinite set using only finite sets?

 

[Dschumanji]

I believe that the answer I am looking for can be found in the post by HallsofIvy in the following thread:

https://www.physicsforums.com/showthread.php?t=205083