Exercises in Analysis by Leszek Gasiński & Nikolaos S. Papageorgiou

Exercises in Analysis by Leszek Gasiński & Nikolaos S. Papageorgiou

Author:Leszek Gasiński & Nikolaos S. Papageorgiou
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham


Passing to a further subsequence if necessary, we may assume that in . Then from Proposition 3.64 and the continuity of h we have

which contradicts (3.108). So, (3.110) holds and from (3.109) and (3.110), we infer that

thus (since is arbitrary) and hence

(3.111)

(see (3.108)). From (3.107) and (3.111), we conclude that

The result fails if we drop the boundedness below by hypothesis. To see this, consider the sequence

Then, is an increasing sequence of lower semicontinuous function and , but .



Download



Copyright Disclaimer:
This site does not store any files on its server. We only index and link to content provided by other sites. Please contact the content providers to delete copyright contents if any and email us, we'll remove relevant links or contents immediately.
ebooksonlinefree.com Copyright 2016 - 2020