proof of Heine-Cantor theorem
Suppose is a compact metric space, continuous on . Let . For each choose such that implies . Note that the collection of balls covers , so by compactness there is a finite subcover, say involving . Take
Then, suppose . By the choice of and the triangle inequality, there exists an such that . Hence,
As were arbitrary, we have that is uniformly continuous.
This proof is similar to one found in Mathematical Principles of Analysis, Rudin.
|Title||proof of Heine-Cantor theorem|
|Date of creation||2013-03-22 15:09:43|
|Last modified on||2013-03-22 15:09:43|
|Last modified by||drini (3)|