2025/04/04

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2025-04-04 17:35:15 +0200 <EvanR> that means I don't have to solve this using the "hint use an indirect proof": Show that family of intervals [a,b] where a^3 <= 2 and 2 <= b^3 is consistent (any interval in the family intersects all the others)
2025-04-04 17:32:39 +0200 <EvanR> cool
2025-04-04 17:32:33 +0200 <EvanR> a < b? yes, then you're done. Or a == b, then a^3==b^3 contradicting a^3 < b^3. Or b < a, using the above, b^3 < a^3, also contradicting.
2025-04-04 17:25:14 +0200 <EvanR> somehow
2025-04-04 17:24:24 +0200 <EvanR> maybe in chapter 17 there will be a thing to prove that any monotonic functions on reals can be downgraded to a monotonic function on rationals
2025-04-04 17:24:00 +0200fp1(~Thunderbi@2001:708:20:1406::1370) (Ping timeout: 272 seconds)
2025-04-04 17:22:15 +0200 <EvanR> again, the task to is to reason from what's assumed and not from "any argument whatever"
2025-04-04 17:21:51 +0200j1n37(~j1n37@user/j1n37) (Ping timeout: 276 seconds)
2025-04-04 17:21:46 +0200 <EvanR> so I can't solve chapter 1 by using chapter 17 from the other book
2025-04-04 17:21:38 +0200j1n37-(~j1n37@user/j1n37) j1n37
2025-04-04 17:21:30 +0200 <EvanR> this is "real anaylsis: a constructive approach"
2025-04-04 17:21:29 +0200 <int-e> (another fact)
2025-04-04 17:21:10 +0200 <int-e> the rationals are a subfield of the reals
2025-04-04 17:20:56 +0200 <EvanR> also it wouldn't even help, are you going to apply a function coded for real numbers to rationals? xD
2025-04-04 17:20:28 +0200notdabs(~Owner@2600:1700:69cf:9000:c0fa:b50a:3031:4dce) (Quit: Leaving)
2025-04-04 17:20:04 +0200 <EvanR> I didn't that's the point
2025-04-04 17:19:56 +0200 <EvanR> but we just proved that
2025-04-04 17:19:56 +0200 <int-e> so how did you get there for the cube root in the reals?
2025-04-04 17:19:36 +0200 <EvanR> it might be but I didn't get there
2025-04-04 17:19:05 +0200int-eshrugs
2025-04-04 17:19:05 +0200 <EvanR> not absolute truths, or "doesn't matter somebody somewhere has already done this one"
2025-04-04 17:18:58 +0200 <int-e> well, f(x) = x^3 being monotonic is a fact about rational number.s
2025-04-04 17:18:32 +0200 <EvanR> I don't understand, I'm not skeptical. I'm trying to go from certain assumptions to a conclusion
2025-04-04 17:16:47 +0200 <int-e> well it's not if you apply the same level of skepticism that you're currently applying to the strict monotonicity of f(x) = x^3 in the rationals
2025-04-04 17:16:12 +0200 <EvanR> well that's a fact about real numbers right
2025-04-04 17:16:08 +0200__jmcantrell__(~weechat@user/jmcantrell) jmcantrell
2025-04-04 17:15:31 +0200 <int-e> TBH the funny part of this is that you were (apparently) willing to accept that the cube root function is strictly monotonic.
2025-04-04 17:13:34 +0200dolio(~dolio@130.44.140.168) dolio
2025-04-04 17:13:18 +0200rit(~rit@2409:40e0:39:317c:5ce4:875:e0a7:d628) (Remote host closed the connection)
2025-04-04 17:13:03 +0200EvanRrecapitulates
2025-04-04 17:12:35 +0200 <EvanR> er
2025-04-04 17:12:30 +0200dolio(~dolio@130.44.140.168) (Client Quit)
2025-04-04 17:12:27 +0200 <int-e> it's just <= because a = 0 is a possibility
2025-04-04 17:12:27 +0200 <EvanR> that's ab < ab
2025-04-04 17:12:05 +0200 <int-e> you multiply a < b by a from the left and by b from the right
2025-04-04 17:11:38 +0200 <EvanR> how did you get a^2b < ab^2
2025-04-04 17:11:29 +0200dolio(~dolio@130.44.140.168) dolio
2025-04-04 17:11:16 +0200 <EvanR> yes I got a^3 < a^2b and ab^2 < b^3 separately
2025-04-04 17:10:52 +0200 <EvanR> wait
2025-04-04 17:10:40 +0200 <int-e> by the laws of an ordered field
2025-04-04 17:10:26 +0200 <int-e> if 0 <= a < b then a^3 <= a^2b <= ab^2 < b^3. The case a < b <= 0 is similar, and a <= 0 <= b and a < b falls to looking at signs
2025-04-04 17:09:31 +0200 <EvanR> how
2025-04-04 17:09:25 +0200 <EvanR> you went from a < b to a^3 < b^3
2025-04-04 17:09:09 +0200 <EvanR> but also you'd have to dispatch the no, actually b < a case
2025-04-04 17:09:07 +0200 <int-e> comparing a and b you get three cases: a < b and a^3 < b^3; a = b an a^3 = b^3; a > b and a^3 > b^3.
2025-04-04 17:08:47 +0200 <EvanR> you use "if a < b then a^3 < b^3" but I couldn't prove this xD
2025-04-04 17:08:39 +0200TheCoffeMaker(~TheCoffeM@user/thecoffemaker) (Ping timeout: 260 seconds)
2025-04-04 17:08:32 +0200 <EvanR> alright case analysis on a < b
2025-04-04 17:07:58 +0200TheCoffeMaker_(~TheCoffeM@186.136.173.70)
2025-04-04 17:07:57 +0200dolio(~dolio@130.44.140.168) (Quit: ZNC 1.9.1 - https://znc.in)