I am looking for the most compact proof for the Bolzano–Weierstrass theorem,anybody have any links/ideas. Thanks.

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- April 29th 2011, 07:40 AMMathSuckerBolzano–Weierstrass theorem
I am looking for the most compact proof for the Bolzano–Weierstrass theorem,anybody have any links/ideas. Thanks.

- April 29th 2011, 07:50 AMPlato
Some statements of this theorem vary. But here is one source.

- April 29th 2011, 10:36 AMMathSucker
What do you think of this one? I want to be able to write it out in the shortest time possible. Is there anything I can trim out?

If is a sequence in the closed segment then there exists a subsequence of that converges to a limit in .

Let be an arbitrary point in , then can contain and infinite number of members of and it can contain a finite number of members of .

Let: contains a finite number of members of

If is in , then has a finite number of members of , and any subsest of can only contain a finite number of members of . Hence, for any , all the points between and are also in .

Let and assume .

Take an arbitrarily small . is not in since is greater than . is in since is less than .

is a subset of . The larger set contains an infinite number of members of and the subset contains a finite number of members of . So, the complement of the subset contains an infinite number of members of .

Thus, every in , has a neighbourhood that contains an infinite number of members of .

Set . Take any member of in and make this the first member of a sequence.

Set . Take any member of in with a higher index than the previous member, and make this the next member of the sequence.

Set and so forth indefinitely.

This results in a subsequence of that converges to . - May 2nd 2011, 04:38 PMHallsofIvy
The best proof based on what? It is, in fact, possible to take "Bolzano-Weierstrasse" as a defining axiom for the real numbers and then the other basic properties (monotone convergence, upper bound property, Heine-Borel, etc.) can be proved from that. But you can also prove "Bolzano-Weierstrasse" from any of them.

- May 2nd 2011, 04:44 PMDrexel28
There are very short proofs, but they may use machinery/terms above the level of your current course. What level is this at?

- May 3rd 2011, 10:55 AMMathSucker
Undergraduate level, but we're given zero worked solutions. The prof basically just writes out proofs on the board, hands us a list of questions, and then walks out. I was just looking for a proof for the "Bolzano-Weierstrasse" theorem that I could bang out on the exam in as little time possible. I found that one, and then wrote it out in my own words and was wondering if it made sense.