Difference between revisions of "10-327/Classnotes for Thursday November 4"

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** If you could find a ball of radius 7 around <math>x</math> which fits inside some set <math>U</math>, and you move <math>x</math> just a 1 unit away to <math>y</math>, then by the triangle inequality the ball of radius 6 around <math>y</math> is entirely contained inside the ball of radius 7 around <math>x</math> so it is entirely contained in <math>U</math>. [[User:Drorbn|Drorbn]] 18:37, 6 November 2010 (EDT)
 
** If you could find a ball of radius 7 around <math>x</math> which fits inside some set <math>U</math>, and you move <math>x</math> just a 1 unit away to <math>y</math>, then by the triangle inequality the ball of radius 6 around <math>y</math> is entirely contained inside the ball of radius 7 around <math>x</math> so it is entirely contained in <math>U</math>. [[User:Drorbn|Drorbn]] 18:37, 6 November 2010 (EDT)
 
***I have some doubts with Lebesgue number lemma too.. this delta(x) isn't a radius that we can fit a ball inside one of the U's. It is the supremum of all possible radius. Wouldn't that give us a problem? Don't we need to subtract some small positive value and then find a valid radius? I am not sure if there should be some technicality involved here.-Kai
 
***I have some doubts with Lebesgue number lemma too.. this delta(x) isn't a radius that we can fit a ball inside one of the U's. It is the supremum of all possible radius. Wouldn't that give us a problem? Don't we need to subtract some small positive value and then find a valid radius? I am not sure if there should be some technicality involved here.-Kai
 +
And once we found this delta(x_0) we should divide by 2 so that delta(x)>delta(x_0) for all x in X? Because it is not necessarily true that we can find a ball of radius delta(x_0) around x_0 to fit into ball?-Kai

Revision as of 01:15, 18 December 2010

See some blackboard shots at BBS/10_327-101104-142342.jpg.

Dror's notes above / Student's notes below
  • Question: Regarding the proof of the Lebesque Number Lemma, let T(x) be the function we were working with in the proof. I am confused with how we reached the conclusion that if d(x,y)<E, then T(y)>= T(x) - E. I know that it was said that this is just an application of the triangle inequality, but I am having a bit of trouble seeing that. Hopefully someone can make this point a bit clearer for me. Thanks! Jason.
    • If you could find a ball of radius 7 around x which fits inside some set U, and you move x just a 1 unit away to y, then by the triangle inequality the ball of radius 6 around y is entirely contained inside the ball of radius 7 around x so it is entirely contained in U. Drorbn 18:37, 6 November 2010 (EDT)
      • I have some doubts with Lebesgue number lemma too.. this delta(x) isn't a radius that we can fit a ball inside one of the U's. It is the supremum of all possible radius. Wouldn't that give us a problem? Don't we need to subtract some small positive value and then find a valid radius? I am not sure if there should be some technicality involved here.-Kai

And once we found this delta(x_0) we should divide by 2 so that delta(x)>delta(x_0) for all x in X? Because it is not necessarily true that we can find a ball of radius delta(x_0) around x_0 to fit into ball?-Kai