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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 +0200 | fp1 | (~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 +0200 | j1n37 | (~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 +0200 | j1n37- | (~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 +0200 | notdabs | (~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 +0200 | int-e | shrugs |
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 +0200 | dolio | (~dolio@130.44.140.168) dolio |
2025-04-04 17:13:18 +0200 | rit | (~rit@2409:40e0:39:317c:5ce4:875:e0a7:d628) (Remote host closed the connection) |
2025-04-04 17:13:03 +0200 | EvanR | recapitulates |
2025-04-04 17:12:35 +0200 | <EvanR> | er |
2025-04-04 17:12:30 +0200 | dolio | (~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 +0200 | dolio | (~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 +0200 | TheCoffeMaker | (~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 +0200 | TheCoffeMaker_ | (~TheCoffeM@186.136.173.70) |
2025-04-04 17:07:57 +0200 | dolio | (~dolio@130.44.140.168) (Quit: ZNC 1.9.1 - https://znc.in) |