Understanding the Weierstrass- Bolzano theorem

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In summary, the conversation discusses the bounded infinite sequence ##a_n=\dfrac{1}{n}## and its subsequence ##b_n= \dfrac{1}{100},\dfrac{1}{200}, \dfrac{1}{300},...## which also tends to ##0##. The question is whether there is another limit point for the sequence, which the author concludes using a given theorem. Additionally, the author claims that the subsequence ##{-1,-1,-1,...}## converges to ##-1##, which should be obvious and can be proven using the definition of a limit.
  • #1
chwala
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I am currently looking at this theorem...basically it states that if you have a given sequence that is bounded and infinite then there exists atleast one limit point. Looking at the attached...i would like to know how the author concludes that the subsequence;

##{-1,-1,-1,...}## converges to ##-1##.

I guess that should follow from previous step...unless there is a mistake.


Find link here;https://math.libretexts.org/Bookshelves/Analysis/Book%3A_Real_Analysis_(Boman_and_Rogers)/07%3A_Inter
1666813869721.png
 
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Let me pick a sequence, say ##a_n=\dfrac{1}{n}## for example...this is a bounded infinite sequence. I can let my subsequence be defined by;

##b_n= \dfrac{1}{100},\dfrac{1}{200}, \dfrac{1}{300},...## the limit of this sequence will still tend to ##0##.

I am looking at it on the aspect of at least ##1## limit point. Question is, is there another limit point for the given sequence? using the given theorem?
 
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chwala said:
i would like to know how the author concludes that the subsequence;

##{-1,-1,-1,...}## converges to ##-1##.

As a standalone claim, this should be exceedingly obvious. If not, apply the definition of a limit without any bells or whistles.
 
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Related to Understanding the Weierstrass- Bolzano theorem

1. What is the Weierstrass-Bolzano theorem?

The Weierstrass-Bolzano theorem, also known as the Bolzano-Weierstrass theorem, is a fundamental theorem in calculus and real analysis. It states that any bounded sequence of real numbers has a convergent subsequence. In simpler terms, if a sequence of numbers is squeezed between two fixed values, then there must be a point in the sequence that converges to a specific value.

2. Who were Weierstrass and Bolzano?

Karl Weierstrass and Bernard Bolzano were both mathematicians who made significant contributions to the development of calculus and real analysis. Weierstrass is best known for his work on the foundations of calculus and for introducing the concept of a limit. Bolzano is known for his work on the foundations of real numbers and for his proof of the intermediate value theorem.

3. How is the Weierstrass-Bolzano theorem used in mathematics?

The Weierstrass-Bolzano theorem is used in many areas of mathematics, including calculus, real analysis, and topology. It is a fundamental tool for proving the convergence of sequences and series, and it is also used to prove the existence of solutions to differential equations and other mathematical problems.

4. Can you give an example of how the Weierstrass-Bolzano theorem is applied?

Sure, suppose we have a sequence of numbers: 1, 1/2, 1/3, 1/4, 1/5, ... This sequence is bounded between 0 and 1, and it is clear that as we continue the sequence, the numbers get closer and closer to 0. By the Weierstrass-Bolzano theorem, we can conclude that there is a subsequence of this sequence that converges to 0.

5. Is the Weierstrass-Bolzano theorem always true?

Yes, the Weierstrass-Bolzano theorem is a fundamental theorem in mathematics and has been proven to be true. However, it is important to note that the theorem only applies to bounded sequences of real numbers. If a sequence is unbounded, the theorem does not hold. Additionally, the theorem only guarantees the existence of a convergent subsequence, not the convergence of the entire sequence.

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