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 .
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Algebraic Geometry | Analytic Geometry |
Differential Geometry | Non-Euclidean Geometries |
Topology |
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