Timeline for Converse to Banach’s fixed point theorem for ordered fields?
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
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| May 26, 2011 at 14:19 | vote | accept | James Propp | ||
| May 26, 2011 at 14:18 | comment | added | James Propp | Thanks, George! I'm marking this question as closed (though if anyone finds a different solution please post that as well). I'll take this opportunity to repeat my request from mathoverflow.net/questions/62340/… : Can anyone point me to a reasonably comprehensive article (or book chapter) explaining which basic theorems of calculus are equivalent to the completeness axiom of the reals and which ones aren't? I'm writing something of my own on this subject and I feel that there must be authors I should acknowledge. | |
| May 25, 2011 at 21:18 | history | edited | George Lowther | CC BY-SA 3.0 |
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| May 25, 2011 at 19:13 | history | edited | George Lowther | CC BY-SA 3.0 |
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| May 25, 2011 at 16:46 | comment | added | George Lowther | That's true for any increasing Cauchy sequence in an ordered field. I'll add some clarifications when I log on later. | |
| May 25, 2011 at 14:38 | comment | added | James Propp | Thanks! I follow most of this, but I don't get the part about "by further passing to a subsequence, we can suppose that $x_{n+2}−x_{n+1} \leq \frac12 (x_{n+1}−x_n)$" (from the third sentence of the first paragraph of the proof of (2)). I know how I'd extract such a subsequence if I were working in the real numbers, by using the existence (and the value) of $\lim x_n$. But I can't use my method here, and trying to do an approximate version leads me into a swamp. Is there an easy way to see why your assertion is true in any ordered ring? | |
| May 25, 2011 at 0:34 | history | edited | George Lowther | CC BY-SA 3.0 |
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| May 25, 2011 at 0:27 | history | edited | George Lowther | CC BY-SA 3.0 |
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| May 25, 2011 at 0:18 | history | edited | George Lowther | CC BY-SA 3.0 |
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| May 24, 2011 at 23:40 | history | answered | George Lowther | CC BY-SA 3.0 |