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Bolzano-Weierstrass

Anonymous Fri 17 Apr 16:37:06 2020 No.11574585 View Reply Original Report

Quoted By: >>11574622

Can someone help explain what the Bolzano-Weierstrass theorem really means?

Thank you

Anonymous

Anonymous Fri 17 Apr 2020 16:46:20 No.11574602 Report

Quoted By:

if you come up with a countably infinite list of real numbers belonging to a closed bounded set then they have to accumulate somewhere, because the real numbers don't have any holes

if you choose a specific order (a subsequence) then you can make them converge

Anonymous

Anonymous Fri 17 Apr 2020 16:52:01 No.11574622 Report

Quoted By: >>11574679

>>11574585
my teacher called it the lion's theorem
so, the lion is a point with infinite points near it (an accumulation point)
If you have a bounded set with infinite cardinality then you must have a lion somewhere.
You find its location by cutting your set into pieces. Some will have finite cardinality, some must not. You focus on those.
and continue cutting forever.

OP here

OP here Fri 17 Apr 2020 17:08:19 No.11574679 Report

Quoted By:

>>11574622
So the lion is the limit of the subsequences of {An}
{An} is bounded by -M and M
The subsequences are taken on the interval [-M,0] or [0,M]
and every infinite subsequence converges to that limit?

How is this proof used in establishing calculus? The fact that the subsequences have a limit means they're differentiable?

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