Baby Rudin question
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From: Tonico on 15 Jul 2010 17:20 On Jul 15, 10:12 pm, master1729 <tommy1...(a)gmail.com> wrote: > David Ullrich wrote : > > > On Tue, 22 Jun 2010 16:59:34 -0700 (PDT), guyc > > <guy.corrig...(a)gmail.com> wrote: > > > >Let A be the set of all positive rationals p such > > that p^2>2. > > > >Let B be the set of all positive rationals p such > > that p^2<2. > > remember david , A and B are sets of RATIONALS. > > > > > >In order to show that A has no infimum, B no > > supremum, Rudin considers > > >the number q = p - (p^2 - )/(p +2) > > > What? A certainly has an inf, and B has a sup! > > ohhh david forgot it ! > > A has no inf and B has no sup , just as RUDIN PROVED. > > > > > At least if we're allowing real numbers. > > AGAIN DAVID , remember the key word , RATIONAL. > > RRRRRAAAATTTTIONAAAALLL. > > > A has no > > _rational_ inf, > > but that can't be what Rudin't proving here, since > > the > > detail below is irrelevant to that. > > sigh. > > > > > Surely what Rudin says he's proving is that A has no > > smallest > > element and B has no largest element? > > you have trouble with rudin hmm. +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ ++++++++++ Tommy, Tommy, Tommy...ts,ts,ts: once again you succeed big time in making a fool of yourself: David is right, as one reading the book can check at once (Baby Rudin = Principles of Mathematical Analysis, page 2, example 1.1). Here ,Rudin writes EXACTLY: "We shall show that A hcontains no largest number and B contains no smallest." It's not about having problems with Rudin buit about UNDERSTANDING the book and knowing a VERY LITTLE mathematics. +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ ++++++++++ > > > The difference > > between "inf" and "smallest element" is hugely > > important. > > thats new. plz tell us , david , what is the ( important ) difference between inf and smallest element. +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ It's almost unbelievable how you have no sense of self shame and decency at all: how dare you try to criticize, and even mock, a MATHEMATICIAN who is right about what he' writing in this matter, wheras you don't know even the above basic difference... A pity, in fact. Tonio +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ > and plz tell us ; > > what is the inf of A in " Let A be the set of all positive rationals p such that p^2>2 " > > and what is the smallest element of A in " Let A be the set of all positive rationals p such that p^2>2 " > > dont forget > > RRAAATTIIIOOONNAAALLLLSSSS > > FRACTIONS if you understand that better. > > your really amazing. > > pulling a ' real ' solution out of a set with rationals. > > is that how you got your phd ? > > maybe we should abolish fractions as DeTurck asked , so that people like you cant be confused anymore. > > and also abolish sets of reals for that matter. > > otherwise you might confuse reals and rationals as you did here. > > > > > >Note that Bryant in the same proof considers the > > number q = p/2 +1/p > > > >Very clever - but what is the motivation for these > > q? Where did they > > >come from? Lots of trial and error? A flash of > > inspiration? > > inspiration ? good one lol > > regards > > the master > > tommy1729
From: David C. Ullrich on 16 Jul 2010 09:59 On Thu, 15 Jul 2010 15:12:12 EDT, master1729 <tommy1729(a)gmail.com> wrote: >David Ullrich wrote : > >> On Tue, 22 Jun 2010 16:59:34 -0700 (PDT), guyc >> <guy.corrigall(a)gmail.com> wrote: >> >> >Let A be the set of all positive rationals p such >> that p^2>2. >> > >> >Let B be the set of all positive rationals p such >> that p^2<2. > >remember david , A and B are sets of RATIONALS. I remember that. >> > >> >In order to show that A has no infimum, B no >> supremum, Rudin considers >> >the number q = p - (p^2 - )/(p +2) >> >> What? A certainly has an inf, and B has a sup! > >ohhh david forgot it ! > >A has no inf and B has no sup , just as RUDIN PROVED. > >> >> At least if we're allowing real numbers. > >AGAIN DAVID , remember the key word , RATIONAL. > >RRRRRAAAATTTTIONAAAALLL. Remember, Tommy, rationals are reals. A does have an inf in the reals, and B does have a sup. >> A has no >> _rational_ inf, >> but that can't be what Rudin't proving here, since >> the >> detail below is irrelevant to that. > >sigh. > >> >> Surely what Rudin says he's proving is that A has no >> smallest >> element and B has no largest element? > > >you have trouble with rudin hmm. Giggle. > >> The difference >> between "inf" and "smallest element" is hugely >> important. > >thats new. plz tell us , david , what is the ( important ) difference between inf and smallest element. That's "new" if you know nothing about analysis. B does not have a smallest element, but it _does_ have an inf (in the reals). The inf of a set is the largest lower bound; it need not be an element of the set. The largest element of a set, on the other hand (if it exists) _is_ an element of the set. The difference is incredibly important. >and plz tell us ; > >what is the inf of A in " Let A be the set of all positive rationals p such that p^2>2 " The inf of A is 0 which is largely irrelevant; more interesting is the fact that the sup of A is sqrt(2). Yes, I understand that sqrt(2) is not rational. >and what is the smallest element of A in " Let A be the set of all positive rationals p such that p^2>2 " A has no smallest element. >dont forget > >RRAAATTIIIOOONNAAALLLLSSSS > >FRACTIONS if you understand that better. > > > >your really amazing. > >pulling a ' real ' solution out of a set with rationals. Huh? Sets do not have "solutions", problems have solutions. The solution to the problem "What is the sup of A" is "sqrt(2)". The fact that sqrt(2) is irrational doesn't matter - it seems to matter if one is confused about the difference between "sup" and "largest element". >is that how you got your phd ? Of course not. I got my PhD by making a fool of myself on sci.math. How you coming with yours? >maybe we should abolish fractions as DeTurck asked , so that people like you cant be confused anymore. > >and also abolish sets of reals for that matter. > >otherwise you might confuse reals and rationals as you did here. > > >> >> >Note that Bryant in the same proof considers the >> number q = p/2 +1/p >> > >> >Very clever - but what is the motivation for these >> q? Where did they >> >come from? Lots of trial and error? A flash of >> inspiration? >> > >inspiration ? good one lol > >regards > >the master > >tommy1729
From: Ki Song on 16 Jul 2010 10:26
Hi Tommy, It looks like you are confused. On Jul 15, 3:12 pm, master1729 <tommy1...(a)gmail.com> wrote: > David Ullrich wrote : > > > On Tue, 22 Jun 2010 16:59:34 -0700 (PDT), guyc > > <guy.corrig...(a)gmail.com> wrote: > > > >Let A be the set of all positive rationals p such > > that p^2>2. > > > >Let B be the set of all positive rationals p such > > that p^2<2. > > remember david , A and B are sets of RATIONALS. > > > > > >In order to show that A has no infimum, B no > > supremum, Rudin considers > > >the number q = p - (p^2 - )/(p +2) > > > What? A certainly has an inf, and B has a sup! > > ohhh david forgot it ! A, B are subsets of rational numbers, which is a subset of the real numbers. inf A and sup B exist in the real numbers. > > A has no inf and B has no sup , just as RUDIN PROVED. > > > > > At least if we're allowing real numbers. > > AGAIN DAVID , remember the key word , RATIONAL. > > RRRRRAAAATTTTIONAAAALLL. What are you, five? > > > A has no > > _rational_ inf, > > but that can't be what Rudin't proving here, since > > the > > detail below is irrelevant to that. > > sigh. > > > > > Surely what Rudin says he's proving is that A has no > > smallest > > element and B has no largest element? > > you have trouble with rudin hmm. > > > The difference > > between "inf" and "smallest element" is hugely > > important. > > thats new. plz tell us , david , what is the ( important ) difference between inf and smallest element. > Do you understand the difference between a infimum and a minimum? Here is a simple example: Let S be the half-open interval (0,1] as a subset of the real numbers. inf S = 0 min S = Does not exist sup S = 1 max S = 1 In the case where the largest/smallest element exists, they are the sup/inf. However, the "smallest" element does not always exist, which is the point. > and plz tell us ; > > what is the inf of A in " Let A be the set of all positive rationals p such that p^2>2 " > > and what is the smallest element of A in " Let A be the set of all positive rationals p such that p^2>2 " > > dont forget > > RRAAATTIIIOOONNAAALLLLSSSS > > FRACTIONS if you understand that better. By fractions, I'm assuming you mean ratios of integers (whose denominator is nonzero). Because fractions can be made of irrational numbers. > > your really amazing. > > pulling a ' real ' solution out of a set with rationals. > > is that how you got your phd ? > > maybe we should abolish fractions as DeTurck asked , so that people like you cant be confused anymore. > > and also abolish sets of reals for that matter. > > otherwise you might confuse reals and rationals as you did here. > I think you are stupid. > > > > >Note that Bryant in the same proof considers the > > number q = p/2 +1/p > > > >Very clever - but what is the motivation for these > > q? Where did they > > >come from? Lots of trial and error? A flash of > > inspiration? > > inspiration ? good one lol > > regards > > the master > > tommy1729 |