What Confuses You About the Bolzano-Weierstrass Theorem Examples?

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SUMMARY

The Bolzano-Weierstrass Theorem states that a subset S of R^n is compact if and only if every sequence in S has a convergent subsequence whose limit is also in S. The discussion highlights confusion regarding the necessity of two cases in the theorem's application, specifically addressing the differences between these cases. The first case involves sequences concentrated on a single segment, while the second case pertains to sequences spread across multiple lines, with the limit converging to (0,0). Visual aids, such as geometric representations, are recommended for better understanding.

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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.
http://www.geocities.com/asdfasdf23135/advcal3.JPG

I am completely completely lost when reading this example.

1. Why do we need 2 cases?

2. How are the 2 cases different?

3. For the second case, how come the subscripts of x_n_j and L_i_j are different? (n and i)

4. I don't understand the solution at all, can somebody please explain it step-by-step what is happening?

I really want to understand this example! Thanks a lot!
 
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Did you draw a picture of the geometry? It would really help. You have an infinite number of line segments which are getting shorter and shorter and closer and closer to the origin. And you do need two cases, one where an infinite number of points are on a single segment (in which case the limit might not be (0,0)) and the case where the points are on an infinite number of lines, in which case the limit is (0,0). Though a single sequence may contain both types of subsequences. Draw a picture and think about it again.
 

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