- #1
shinobi20
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BTW. theorem 1.1 just states that Every bounded and monotonic sequence is convergent.
Question about Bolzano-Weierstrass Theorem Proof
In summary, Serge Lang's proof of the Bolzano-Weierstrass Theorem is odd and difficult to understand. He uses different sources to try to understand the theorem, but eventually fails. The theorem states that every bounded and monotonic sequence is convergent, and Lang's proof relies on this fact. However, his proof does not rely on any other information, and is instead based on the assumption that the Xn's will oscillate randomly. If this assumption is not true, then Cn+1 will not be less than Cn.
BTW. theorem 1.1 just states that Every bounded and monotonic sequence is convergent.
- #2
lavinia
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What don't you understand about the proof?
- #3
shinobi20
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I don't understand the proof of the Bolzano-Weierstrass Theorem according to Serge Lang...
- #4
lavinia
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Yes but what about it don't you understand?
- #5
shinobi20
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How can Cn be increasing? How can we be sure that the Xn will less than Xn+1?
- #6
lavinia
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##C_{n}## is the greatest lower bound of the sequence of ##x_{n}##'s except for the first ##n-1## of them. If you remove some more of the ##x_{n} ##'s then the greatest lower bound can not be less than ##C_{n}##.shinobi20 said:
- #7
shinobi20
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Yes but how can we be sure that Cn+1 will not be less than Cn if ever those Xn's oscillate in a very random manner? For example, if Xn+1 is less than Xn, then Cn+1 is the GLB of the set Xn+1's, and Cn is the GLB of the Xn's but this implies Cn+1 is less than Cn.
- #8
lavinia
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Because they are greatest lower bounds. It doesn't matter if the X's oscillate. ##C_{n}## is lower than all of them except the first ##n-1##. ##C_{n+1}## is lower than all of them except one less so that one removed might be very low.shinobi20 said:
- #9
shinobi20
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Oh! Now I got it, because I was thinking that Cn's can overlap or surpass the Xn+1's and vice versa... Thanks!
FAQ: Question about Bolzano-Weierstrass Theorem Proof
What is the Bolzano-Weierstrass Theorem?
The Bolzano-Weierstrass Theorem is a fundamental theorem in real analysis, named after mathematicians Bernard Bolzano and Karl Weierstrass. It states that if a bounded infinite set of real numbers exists, then there exists at least one point in the set that has a convergent subsequence.
What is the importance of the Bolzano-Weierstrass Theorem?
The Bolzano-Weierstrass Theorem is important because it provides a powerful tool for proving the convergence of sequences in real analysis. It also has applications in areas such as optimization, functional analysis, and differential equations.
How is the Bolzano-Weierstrass Theorem proved?
The Bolzano-Weierstrass Theorem can be proved using various methods, such as the nested interval theorem, the bisection method, or the completeness axiom. However, the most common proof is known as the "diagonal argument" which involves dividing the set into two subsets and then recursively choosing subsequences from each subset until a convergent subsequence is found.
Can the Bolzano-Weierstrass Theorem be extended to higher dimensions?
Yes, the Bolzano-Weierstrass Theorem can be extended to higher dimensions. In fact, the theorem is often stated in a more general form known as the Bolzano-Weierstrass Theorem for Euclidean spaces, which states that if a bounded infinite set exists in an n-dimensional Euclidean space, then there exists at least one point in the set that has a convergent subsequence.
Are there any limitations to the Bolzano-Weierstrass Theorem?
Yes, there are some limitations to the Bolzano-Weierstrass Theorem. It only applies to bounded sets in real or Euclidean spaces, and it does not guarantee the uniqueness of the convergent subsequence. Additionally, the theorem cannot be applied to unbounded sets or sets in other types of spaces such as complex or metric spaces.