proof of Dini’s theorem
Without loss of generality we will assume that is compact and, by replacing
with , that the net converges
monotonically to 0.
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 |
---|---|
Canonical name | ProofOfDinisTheorem |
Date of creation | 2013-03-22 12:44:13 |
Last modified on | 2013-03-22 12:44:13 |
Owner | mathcam (2727) |
Last modified by | mathcam (2727) |
Numerical id | 5 |
Author | mathcam (2727) |
Entry type | Proof |
Classification | msc 54A20 |