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Analysis math homework

  1. Oct 31, 2009 #1
    1. The problem statement, all variables and given/known data
    Let A [tex]\in[/tex]R be a non-empty,bounded set .Define
    B=A+1=[tex]\left\{[/tex]a+1:a[tex]\in[/tex]A[tex]\right\}[/tex] Prove that sup(B) =sup(A)+1
    2. Relevant equations
    3. The attempt at a solution
    let a[tex]\in[/tex]A b[tex]\in[/tex]B s[tex]\in[/tex]R because B=A+1=[tex]\left\{[/tex]a+1:a[tex]\in[/tex]A[tex]\right\}[/tex]
    so b=a+1 [tex]\forall[/tex]b[tex]\in[/tex]B s>= b [tex]\rightarrow[/tex] s>=a+1 , s>=a so sup(B) =sup(A)+1
    i thought that is too simple in my way , any other ways?
     
  2. jcsd
  3. Oct 31, 2009 #2

    jgens

    User Avatar
    Gold Member

    Re: analysis

    From the definition of [itex]A[/itex] and [itex]B[/itex], if [itex]\mathrm{sup}(A) = \alpha[/itex] then we clearly have that [itex]\alpha + 1 \geq a + 1 = b[/itex]. This proves that [itex]\mathrm{sup}(A) + 1[/itex] is an upperbound for [itex]B[/itex]. Now can you prove that this number must be the least upper bound?
     
  4. Oct 31, 2009 #3
    Re: analysis

    i see ,thx
     
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