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Greatest lower bound problem - Rudin POMA Ch1 Exercise 5

  1. Oct 9, 2012 #1
    1. The problem statement, all variables and given/known data
    5. Let ##A## be a nonempty set of real numbers which is bounded below. Let ##-A## be the set of all numbers ##-x##, where ##x\in A##.Prove that
    $$\inf A=-\sup(-A)\text{.}$$


    2. Relevant equations

    3. The attempt at a solution
    Does the proof below look OK? I am a bit uneasy about just multiplying through (2) by -1 to get (4). Thanks in advance.

    Proof: As ##A## is bounded below ##\inf A## exists, let's call this ##\alpha##. Then we can say
    (1) ##\forall x\in A##, ##x\geq\alpha##, and
    (2) if ##\gamma>\alpha## then ##\exists y\in A## such that ##\alpha\leq y<\gamma## (in other words ##\gamma## is not a lower bound of A).
    To get set ##-A## we must multiply each member of ##A## by ##-1##. So if ##x\geq\alpha## is true then ##-x\leq-\alpha## is true, and as ##-x## is an arbitrary member of ##-A## it must be that
    (3) ##\forall-x\in-A##, ##-x\leq-\alpha##,
    so ##-\alpha## is an upper bound of ##A##. Similarly, multiplying the inequalities in (2) by ##-1## means that
    (4) if ##-\gamma<-\alpha## then ##\exists-y\in-A## such that ##-\alpha\geq-y>-\gamma##,
    so that ##-\alpha## is the least upper bound of ##-A##. Thus we can write ##\sup(-A)=-\alpha=-\inf(A)## or ##-\sup(-A)=\inf(A)##, which is what we needed to show. ##\square##
     
  2. jcsd
  3. Oct 9, 2012 #2

    jbunniii

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    Yes, your proof looks OK. If multiplying (2) by -1 to get (4) makes you uneasy, then try writing it in a bit more detail. For example:

    If [itex]\gamma < -\alpha[/itex], then [itex]\alpha < -\gamma[/itex], so by (2), there exists [itex]y \in A[/itex] such that [itex]\alpha \leq y < -\gamma[/itex]. But [itex]y \in A[/itex] iff [itex]-y \in -A[/itex], and [itex]\alpha \leq y < -\gamma[/itex] iff [itex]\gamma < -y \leq -\alpha[/itex], so this shows that there is an element of [itex]-A[/itex], namely [itex]-y[/itex], which is larger than [itex]\gamma[/itex]. Therefore...
     
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