limit points of uncountable subsets of R^n
Proof. For any let
i.e. is a closed ball centered in with radius . Assume, that for any the set
is finite. Then would be at most countable. Contradiction, since is uncountable. Thus, there exists such that is infinite. But and since is compact (and is infinite), then there exists limit point of in . This completes the proof.
Corollary. If is uncountable, then there exist infinitely many limit points of in .
Proof. Assume, that there are finitely many limit points of , namely . For define
Briefly speaking, is a complement of a union of closed balls centered at with radii . Of course since there are finitely many limit points. Let
Assume, that is countable for every . Then
for some . Thus is different from any . Contradiction, since is also a limit point of .
|Title||limit points of uncountable subsets of R^n|
|Date of creation||2013-03-22 19:07:57|
|Last modified on||2013-03-22 19:07:57|
|Last modified by||joking (16130)|