Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

    micromass

    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

    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

    micromass

    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

    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

    micromass

    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

    Re: supremum

    That's indeed correct.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Equivalent definition of the supremum
  1. Proof with supremum. (Replies: 11)

Loading...