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Equivalent definition of the supremum

  1. Feb 3, 2012 #1
    Hello everyone,

    is the following an equivalent definition of the supremum of a set M, M subset of R?

    y=sup{M} if and only if

    given that y is an upper bound of M and x is any real number,
    y >= x implies there exists m in M so that m >=x.

    pf:
    Let x_n be a sequence approaching y from the right. Then
    for each x_n, there exists m_n in M so that m_n >=x_n.
    Since y is an upper bound of M, then we have that y= lim m_n >= lim x_n.
    Therefore, if m' is any another upper bound, then m'>=y for all m in M.

    Thanks
     
  2. jcsd
  3. Feb 3, 2012 #2
    Re: supremum

    This is not true. Specifically, if y=sup(M), then it does not need to holds that y>=x implies m>=x for an m.

    Indeed, take y=x.
     
  4. Feb 4, 2012 #3
    Re: supremum

    If you take y=x, then there exists m in M so that m>=x=y. But y is an upper bound of M, so y=x=m.
     
  5. Feb 4, 2012 #4
    Re: supremum

    Take A=]0,1[, then y=1 is a supremum. Does there exist an m in A such that m>=y??
     
  6. Feb 4, 2012 #5
    Re: supremum

    What if we replace it by

    y=sup{M} if and only if

    given that y is an upper bound of M and x is any real number,
    y >x implies there exists m in M so that m >=x.

    Thanks
     
  7. Feb 4, 2012 #6
    Re: supremum

    That's indeed correct.
     
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