Two other proofs of the Bolzano-Weierstrass Theorem. We prove the result: If $ \ mathbb{X} = \{x_n: n \in \mathbb is a sequence of real numbers. Theorem. (Bolzano-Weierstrass). Every bounded sequence has a convergent subsequence. proof: Let be a bounded sequence. Then, there exists an interval. The proof doesn’t assume that one of the half-intervals has infinitely many terms while the other has finitely many terms; it only says that at least one of the halves .

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It doesn’t matter, but it’s a neater proof to say “choose the left hand one. I know because otherwise you wouldn’t have thought to ask this question. I just weierdtrass convince myself to accept this part. Views Read Edit View history. Your brain does a very good job of checking the details.

### The Bolzano-Weierstrass Theorem – Mathonline

Sign up or log in Sign up using Google. Bolzxno proof is from the book Advanced Calculus: In mathematics, specifically in real analysisthe Bolzano—Weierstrass theoremnamed after Bernard Bolzano and Karl Weierstrassis a fundamental result about convergence in a finite-dimensional Euclidean space R n. There are different important equilibrium concepts in economics, the proofs of the existence of which often require variations of the Bolzano—Weierstrass theorem.

The proof doesn’t assume that one of the half-intervals has theorrem many terms while the other has finitely many terms; it only says that at least one of the halves has infinitely many terms, and one can be chosen arbitrarily.

Moreover, A weieratrass be closed, since from a noninterior point x in the complement of Aone can build an A -valued sequence converging to x. The Bolzano—Weierstrass theorem allows pfoof to prove that if the set of allocations is compact and non-empty, then the system has a Pareto-efficient allocation.

Because each sequence has infinitely many members, there must be at least one subinterval which contains infinitely many members. The theorem states that each bounded sequence in R n has a convergent subsequence.

Email Required, but never shown. By clicking “Post Your Answer”, you acknowledge that you have read our updated terms of serviceprivacy policy and cookie policy prof, and that your continued use of the website is subject to these policies. We continue this process infinitely many times. One example is the existence of a Bolzaano efficient allocation.

By using this site, you agree to the Terms of Use and Privacy Policy. Mathematics Stack Exchange works best with JavaScript enabled. Michael M 2, 6 Post Your Answer Discard By clicking “Post Your Answer”, you acknowledge that you have read our updated terms of serviceprivacy policy and cookie policyand that your continued use of the website is subject to these policies.

An allocation is a matrix of consumption bundles for agents in an economy, and an allocation is Pareto efficient if no change can be made to it which makes no agent worse off and at least one agent better off here rows of the allocation matrix must be rankable by a preference relation.

Suppose A is a subset of R n with the property that every sequence in A has a subsequence converging to an element of A. Now, to answer your question, as others have said and you have said yourselfit’s entirely possible that both intervals have infinitely many elements from the sequence in them.

Home Questions Tags Users Unanswered. Sign up using Email and Password. Because we halve the length of an interval at each step the limit of the interval’s length is zero.

It follows from the monotone convergence theorem that this subsequence must converge. Help me understand the proof for Bolzano-Weierstrass Theorem!

## Bolzano–Weierstrass theorem

I am now satisfied and convinced, thank you so much for the explanation! In fact, general topology tells us that a metrizable space is compact if and only if it is sequentially compact, theoreem that the Bolzano—Weierstrass and Heine—Borel theorems are essentially the same. Retrieved from ” https: If that’s the case, you can pick either one and move on to the next step.

Post as a guest Name. Thanks it makes sense now! Theorems in real analysis Compactness theorems. Indeed, we have the following result. There is also an alternative proof of the Bolzano—Weierstrass theorem using nested prolf. To show existence, you bolzan have to show you can find one. It has since become an essential theorem of analysis. From Wikipedia, the free encyclopedia.

Some fifty years later the result was identified as significant in its own right, and proved again by Weierstrass. By using our site, you acknowledge that you have read and understand our Cookie PolicyPrivacy Policyand our Terms of Service.

Since you can choose either one in this case, why not always just choose the left hand one?

This page was last edited on 20 Novemberat We will use the method of interval-halving introduced previously to prove the existence of least upper bounds. This form of the theorem makes especially clear the analogy to the Heine—Borel theoremwhich asserts that a subset theorrem R n is compact if and only if it is closed and bounded.

It was actually first proved by Bolzano in as a lemma in the proof of the intermediate value theorem. Does that mean this proof only proves that there is only one subsequence that is convergent?

### Two other proofs of the Bolzano-Weierstrass Theorem

Thank you for the comments! Sign up using Facebook. Thus we get a sequence of nested intervals. I boxed the part I didn’t understand.