proof of Dini’s theorem
Let . For each , we can choose an , such that . Since is continuous, there is an open neighbourhood of , such that for each , we have . The open sets cover , which is compact, so we can choose finitely many such that the also cover . Then, if , we have for each and , since the sequence is monotonically decreasing. Thus, converges to 0 uniformly on , which was to be proven.
|Title||proof of Dini’s theorem|
|Date of creation||2013-03-22 12:44:13|
|Last modified on||2013-03-22 12:44:13|
|Last modified by||mathcam (2727)|