How to Prove Compactness of Infinite Union of Line Segments in R^2?

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Homework Help Overview

The discussion revolves around proving properties of sets in R^n, specifically focusing on the openness of a defined set and the compactness of an infinite union of line segments in R^2. The original poster expresses confusion regarding the application of the Bolzano-Weierstrass Theorem and the distinction between cases in the proof process.

Discussion Character

  • Exploratory, Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants explore the definition of an open set and attempt to prove that a specific set is open. They discuss the use of the triangle inequality and the need to establish bounds for points in the set.
  • There is a focus on the Bolzano-Weierstrass Theorem in relation to proving the compactness of an infinite union of line segments, with participants questioning how to handle sequences of points and the implications of different cases.
  • Some participants seek clarification on the necessity of distinguishing between cases when analyzing sequences of points on line segments.

Discussion Status

The discussion is ongoing, with participants sharing their attempts and insights. Some guidance has been offered regarding the application of theorems and the structure of the proof, but there is no explicit consensus on the approach to take. Participants continue to express confusion and seek further clarification on specific points.

Contextual Notes

Participants mention a lack of examples in their textbook, contributing to their difficulties in applying theorems. There is also a sense of urgency as some participants prepare for an upcoming test, which adds to their frustration.

kingwinner
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1) Prove rigorously that S={(x,y) | 1< x^2 + y^2 <4} in R^n is open using the following definition of an open set:
A set S C R^n is "open" if for all x E S, there exists some r>0 s.t. all y E R^n satisfying |y-x|<r also belongs to S.

[My attempt:
Let x E S, r1 = 2 - |x|, r2= |x| - 1, r = min {r1,r2}
Let y E R^n s.t. |y-x|<r
=>|y-x|<r1 and |y-x|<r2

|y|<=|y-x|+|x| (triangle inequality) [<= means less than or equal to]
<r1+|x|=2-|x|+|x|=2
So |y|<2

Now if I can prove that |y|>1, then I am done (y belongs to S). However, I tried many different ways, but still unable to prove that |y|>1, what should I do? ]


2) Let Li denote the line segment in R^2 from the origin (0,0) to the point (1/i, sqrt(1/i) ) on the curve f(x)=sqrt x. Prove that the infinite union S=U(i=1 to infinity) Li is compact using the Bolzano-Weierstrass Theorem.

Bolzano-Weierstrass Theorem: Let S C R^n. Then S is compact (bounded and closed) iff every sequence of points in S has a convergent subsequence whose limit lies in S.

Any help, explanation, or hints? I am feeling totally blank on this question...


My textbook basically has no examples, so I don't know how to use the theorems and I am really frustrated when doing the exercises...

I hope that someone would be kind enough to help me out! Thank you!
 
Last edited:
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kingwinner said:
1) Prove rigorously that S={(x,y) | 1< x^2 + y^2 <4} in R^n is open using the following definition of an open set:
A set S C R^n is "open" if for all x E S, there exists some r>0 s.t. all y E R^n satisfying |y-x|<r also belongs to S.

[My attempt:
Let x E S, r1 = 2 - |x|, r2= |x| - 1, r = min {r1,r2}
Let y E R^n s.t. |y-x|<r
=>|y-x|<r1 and |y-x|<r2

|y|<=|y-x|+|x| (triangle inequality) [<= means less than or equal to]
<r1+|x|=2-|x|+|x|=2
So |y|<2

Now if I can prove that |y|>1, then I am done (y belongs to S). However, I tried many different ways, but still unable to prove that |y|>1, what should I do? ]
Use the triangle inequality but the other way around.
Since |y- x|= |x- y|< r2, |x|< |x-y|+ |y|, |x|< r2+ |y|, |x|< |x|-1+ |y|, 1< |y|.


2) Let Li denote the line segment in R^n from the origin (0,0) to the point (1/i, sqrt(1/i) ) on the curve f(x)=sqrt x. Prove that the infinite union S=U(i=1 to infinity) Li is compact using the Bolzano-Weierstrass Theorem.

Bolzano-Weierstrass Theorem: Let S C R^n. Then S is compact (bounded and closed) iff every sequence of points in S has a convergent subsequence whose limit lies in S.

Any help, explanation, or hints? I am feeling totally blank on this question...


My textbook basically has no examples, so I don't know how to use the theorems and I am really frustrated when doing the exercises...

I hope that someone would be kind enough to help me out! Thank you!
Well, only the obvious. Since you are told to use the Bolzano-Weierstrasse theorem, Let {xn} be a sequence of points in S. Prove that it has a convergent subsequence! You might want to break it into parts: If there happen to be an infinite number of points on a single one of the line segments Li, can you prove that has a convergent subsequence? (Is each Li closed and bounded?) If not then there must be an infinite subsequnce on {ai} with each ai on a different Li. Can you prove that that sequence converges to 0?
 
2) I actually have a worked example to a similar question, and they did the same as you suggested, but I don't understand the difference between the following 2 cases and I am not sure why we need 2 cases...can you please explain more on this part?

Case 1: If there happen to be an infinite number of points on a single one of the line segments L

Case 2: If not then there must be an infinite subsequnce on {ai} with each ai on a different Li

I have a test tomorrow and I am still stuck on these kinds of questions...

Thanks a lot for your help!
 
Last edited:
kingwinner said:
2) I actually have a worked example to a similar question, and they did the same as you suggested, but I don't understand the difference between the following 2 cases and I am not sure why we need 2 cases...can you please explain more on this part?

Case 1: If there happen to be an infinite number of points on a single one of the line segments L
Yes, that's true. And what is their limit?

Case 2: If not then there must be an infinite subsequnce on {ai} with each ai on a different Li
No, I didn't say that. There must be an infinite sequence {ai} on ONE Li. What is its limit?

I have a test tomorrow and I am still stuck on these kinds of questions...

Thanks a lot for your help!
 
Sorry, for question #2, I am completely lost now...

Can anyone explain this question from scratch and provide the steps to solve this problem? What theorems do I need to use in these steps?

Thanks!
 
I am sure someone here knows how to solve this problem. Please help me...
 

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