Heine-Porel from Bolzano-Weierstrass

[soopo]

Homework Statement

How can you prove Heine-Porel (unit interval is compact) theorem by Bolzano-Weierstrass theorem (there is a limit in a continuous and bounded interval)?

The Attempt at a Solution

Compact means that the sequence is complete and totally bounded.

Unit interval perhaps means a bounded interval of an unit length.

I do not see how the continuity in Bolzano-Weierstrass theorem is related to
Heine-Porel theorem.

 

[lanedance]

Hi soopo - disclaimer first as I'm fairly new to this stuff myself

i think the Heine-Borel theorem actually states that a subsapce of Rⁿ is compact iff it is closed & bounded

definition of compact is for every open cover, there exists a finite subcover
http://en.wikipedia.org/wiki/Open_cover

bolzano theorem states that each bounded sequence in Rⁿ has a convergent subsequence.
http://en.wikipedia.org/wiki/Bolzano-Weierstrass_theorem

so start by assuming you have a closed & bounded set eg. [0,1]. Then try & show its compact... and then the other way.. and how can you relate the Bolzano conditions to the compactness cover definition & can you find a path between them?

by the way you would have a better chance of getting answered on the calculus part of this forum... Dick & HallsofIvy would rip through it

 

[HallsofIvy]

what "continuity" are you talking about? Bolzano-Weierstrasse says that every bounded sequence of real numbers contains a convergent subsequence. That says nothing about "continuity". I don't understand what you mean by "continuous interval"- it might be a language problem- "connected interval"?

Essentially, the proof that "Bolzano-Weierstrasse" implies "Heine-Borel" uses the fact that a sequence in a closed and bounded set (or, specifically, [0, 1]) is bounded because the set is bouded, so, by Bolzano-Weierstrasse, contains a convergent subsequence. The fact that the set is closed the implies that convergent subsequence converges to a point in the set and so Heine-Borel.

 

[soopo]

Essentially, the proof that "Bolzano-Weierstrasse" implies "Heine-Borel" uses the fact that a sequence in a closed and bounded set (or, specifically, [0, 1]) is bounded because the set is bouded, so, by Bolzano-Weierstrasse, contains a convergent subsequence. The fact that the set is closed the implies that convergent subsequence converges to a point in the set and so Heine-Borel.

The reason why B-W theorem implies Heine-Borel is that B-W tells us that a closed and bounded set (eg [0, 1]) contains a convergent subsequence.

I don't understand what you mean by "continuous interval"- it might be a language problem- "connected interval"?

I have been using a continuous interval as a synonym for a connected interval.
It seems that my convention is false.

Thank you for your reply!