Insights Blog
-- Browse All Articles --
Physics Articles
Physics Tutorials
Physics Guides
Physics FAQ
Math Articles
Math Tutorials
Math Guides
Math FAQ
Education Articles
Education Guides
Bio/Chem Articles
Technology Guides
Computer Science Tutorials
Forums
General Math
Calculus
Differential Equations
Topology and Analysis
Linear and Abstract Algebra
Differential Geometry
Set Theory, Logic, Probability, Statistics
MATLAB, Maple, Mathematica, LaTeX
Trending
Featured Threads
Log in
Register
What's new
Search
Search
Search titles only
By:
General Math
Calculus
Differential Equations
Topology and Analysis
Linear and Abstract Algebra
Differential Geometry
Set Theory, Logic, Probability, Statistics
MATLAB, Maple, Mathematica, LaTeX
Menu
Log in
Register
Navigation
More options
Contact us
Close Menu
JavaScript is disabled. For a better experience, please enable JavaScript in your browser before proceeding.
You are using an out of date browser. It may not display this or other websites correctly.
You should upgrade or use an
alternative browser
.
Forums
Mathematics
Calculus
Proving half of the Heine-Borel theorem
Reply to thread
Message
[QUOTE="Office_Shredder, post: 6573967, member: 53426"] This just proves it works for your not very inspired choice. Here's one that's a bit tougher. Remember, your argument is not supposed to care which element of the open cover you pick. ##U## contains all sets of the form ##(-1,1-1/n)##. It also contains ##(0.5,1.5)##. It also contains all sets of the form ##(1.5,3-1/n)##, and lastly contains ##(2.5,3.5)##. Then you start by taking points from all these sets ##(-1,1-1/n)##, and finding a slightly larger point in the set, and adding the next ##(-1,1-1/(n+1))##. After a countable number of steps, you've still only (barely not even) covered [0,1]. Then you get the contradiction step which effectively finds the set ##(0.5,1.5)## and you've shown that [0,1] is covered. You haven't proven that after countable steps you can cover the whole set, because I can give you a cover such that if you pick out the "first" countable collection of open sets, you've still only covered part of the whole space. Obviously in theory this argument can be fixed, but I think it's a mess to do so. Even just on the interval [0,1] I could have picked a cover that only reached out to 1/2 on the first round, then 3/4 on the second round, then 7/8 on the third round etc. So I can force you to do infinitely many rounds of this. How confident are you that this can only be forced countably infinitely many times? It might be true, or maybe it just turns out that this is not a method that is guaranteed to construct a finite cover eventually [/QUOTE]
Insert quotes…
Post reply
Forums
Mathematics
Calculus
Proving half of the Heine-Borel theorem
Back
Top