Rudin 2.41, why does an unbounded infinite set have no limit points?

In summary, Rudin states that if E is not bounded, then E contains points xn with |xn|>n (n = 1, 2, 3,...). However, this contradicts his assertion that S, the set of points greater than 1 and ε>0, has no limit point in ℝk.
  • #1
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Please let me know if this should be in the Homework section.

Part of Rudin 2.41 says that for a set E in ℝk... "If E is not bounded, then E contains points xn with |xn|>n (n = 1, 2, 3,...)." I can understand the argument this far. I do not get the next sentence "The set S consisting of these points xn is infinite and clearly has no limit point in ℝk". I can understand that S is infinite but to me S clearly does have limit points in ℝk. For example, take the real line, n = 1, S to be the set of points greater than 1 and ε>0. Every neighborhood of p = 2 with radius ε would contain a point in S making p a limit point. This contradicts what Rudin states as 'clearly'. What am I missing here?

Thanks,

Ed
 
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  • #2
It's not true that an unbounded infinite set has no limit points.

But it's not an arbitrary such set. It's a sequence, so a countable set. A finite set has no limit points. So, any finite subset has no limit points and it's going off to infinity. If there were a limit point, that would contradict the fact that each x_n is bigger than n because there would be some subsequence that converges. A convergent sequence is bounded.
 
  • #3
EdMel said:
For example, take the real line, n = 1, S to be the set of points greater than 1 and ε>0. Every neighborhood of p = 2 with radius ε would contain a point in S making p a limit point. This contradicts what Rudin states as 'clearly'. What am I missing here?

Thanks,

Ed

Pretty much what the previous poster said, but you're missing the point that rudin is not defining S as ALL points greater than some n, but rather each x_n is ONE point greater than n: in other words, an infinite sequence.

Rudin defines this sequence using distinct points, thus while there are an infinite amount, the points are "discrete" objects, if you will. They do not form a connected set, there is distance between each one. Hence, there is no point such that EVERY neighborhood contains a point of the sequence.
 
  • #4
I get it now. Back in 2.7 he defines a sequence and whenever you see x_n for (n=1, 2, 3,...) you should read this as a sequence.

Thanks.
 

1. What is Rudin 2.41?

Rudin 2.41 is a theorem in mathematical analysis, specifically in the field of real analysis, which states that an unbounded infinite set has no limit points.

2. What is an unbounded infinite set?

An unbounded infinite set is a set that has an infinite number of elements and does not have a maximum or minimum element. In other words, the elements in the set can be infinitely large or small without any limit.

3. What are limit points?

Limit points, also known as accumulation points, are points in a set that can be approached arbitrarily closely by elements in the set. In other words, for any given distance, there exists an element in the set that is within that distance from the limit point.

4. Why does an unbounded infinite set have no limit points?

This is because an unbounded infinite set does not have any elements that are infinitely close to each other. Therefore, it is not possible for any element in the set to approach another element arbitrarily closely, which is a necessary condition for a limit point.

5. How does Rudin 2.41 relate to the concept of unbounded infinite sets?

Rudin 2.41 is a theorem that explains the behavior of unbounded infinite sets. It states that these sets do not have any limit points, which is a unique characteristic of unbounded sets compared to bounded sets. This theorem helps us understand the properties of unbounded infinite sets in real analysis.

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