Marsden, J. E. (1968) Countable and net convergence. American Mathematical Monthly, 75 (4). ISSN 0002-9890. https://resolver.caltech.edu/CaltechAUTHORS:20100824-104216924
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Abstract
It is well known that Lebesgue's dominated convergence theorem does not hold for nets; that is, having a countable sequence is essential. On the other hand, for a real valued function on an interval, sequences do suffice; that is, lim_(x → y)f(x) = a iff lim_(n → ∞)f(x_n) = a for every sequence x_n → y. The purpose of this note is to isolate the basic reasons for these phenomena.
| Item Type: | Article | ||||||
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| Additional Information: | © 1968. Reprinted from the American Mathematical Monthly. | ||||||
| Issue or Number: | 4 | ||||||
| Record Number: | CaltechAUTHORS:20100824-104216924 | ||||||
| Persistent URL: | https://resolver.caltech.edu/CaltechAUTHORS:20100824-104216924 | ||||||
| Usage Policy: | No commercial reproduction, distribution, display or performance rights in this work are provided. | ||||||
| ID Code: | 19628 | ||||||
| Collection: | CaltechAUTHORS | ||||||
| Deposited By: | Ruth Sustaita | ||||||
| Deposited On: | 09 Sep 2010 16:48 | ||||||
| Last Modified: | 03 Oct 2019 01:59 |
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