What Confuses You About the Bolzano-Weierstrass Theorem Examples?

  • Thread starter kingwinner
  • Start date
  • Tags
    Theorem
In summary, the Bolzano-Weierstrass Theorem states that a set S in R^n is compact if and only if every sequence of points in S has a convergent subsequence whose limit lies in S. This theorem is applicable in two cases, one where an infinite number of points are on a single segment and the other where the points are on an infinite number of lines, with different subscripts denoting the difference between the two cases. Drawing a picture can help in understanding the concept better.
  • #1
kingwinner
1,270
0
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!
 
Physics news on Phys.org
  • #2
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.
 

FAQ: What Confuses You About the Bolzano-Weierstrass Theorem Examples?

What is the Bolzano-Weierstrass Theorem?

The Bolzano-Weierstrass Theorem, also known as the Bolzano-Cauchy Theorem, is a fundamental theorem in mathematics that states that every bounded sequence in Euclidean space has a convergent subsequence. It is named after the mathematicians Bernard Bolzano and Karl Weierstrass, who independently proved the theorem in the 19th century.

What does the Bolzano-Weierstrass Theorem imply?

The Bolzano-Weierstrass Theorem implies that any bounded sequence in Euclidean space has at least one point of accumulation, which is a point where an infinite number of terms of the sequence are arbitrarily close to. This is a powerful result that has many applications in analysis and other areas of mathematics.

How does the Bolzano-Weierstrass Theorem relate to the Intermediate Value Theorem?

The Bolzano-Weierstrass Theorem is closely related to the Intermediate Value Theorem, which states that if a continuous function has different signs at its endpoints, then it must have at least one root in between. This is because the Bolzano-Weierstrass Theorem can be used to prove the Intermediate Value Theorem for continuous functions defined on a closed interval.

Is the Bolzano-Weierstrass Theorem applicable to infinite sequences?

Yes, the Bolzano-Weierstrass Theorem is applicable to both finite and infinite sequences. It guarantees the existence of a convergent subsequence for any bounded sequence, regardless of its length. This makes it a powerful tool for analyzing the behavior of infinite sequences in mathematics.

Can the Bolzano-Weierstrass Theorem be generalized to higher dimensions?

Yes, the Bolzano-Weierstrass Theorem can be generalized to higher dimensions. In fact, the theorem was originally proven in n-dimensional Euclidean space, where n is any positive integer. This generalization is known as the Bolzano-Weierstrass Theorem in n-dimensional space.

Similar threads

Can't understand this Bolzano-Weierstrass proof
Replies
8
Views
2K
What Are Subsequences in the Bolzano–Weierstrass Theorem Proof?
Replies
9
Views
5K
Real Analysis Question: Sequences and Closed Sets
Replies
3
Views
2K
What does the bolded part in the proof of the Bolzano-Weierstrass theorem mean?
Replies
8
Views
5K
Does a Bounded, Divergent Sequence Always Have Multiple Convergent Subsequences?
Replies
8
Views
3K
Question about the Bolzano Weierstrass analogue in Hilbert spaces
Replies
2
Views
3K
How to Prove Compactness of Infinite Union of Line Segments in R^2?
Replies
5
Views
3K
MHB Why Is the Function Bounded in the Extreme Value Theorem Proof?
Replies
1
Views
2K
Divergence theorem example question (Thomas' Calculus)
Replies
2
Views
3K
Question about Bolzano-Weierstrass Theorem Proof
Replies
8
Views
4K

Hot Threads

Recent Insights

Back
Top