Results 1 to 2 of 2

Math Help - Supremum Bounds

  1. #1
    Junior Member
    Joined
    Sep 2007
    Posts
    46

    Supremum Bounds

    How can there be more than 1 upper bound? Can't you use the same argument for showing that there is only 1 greatest lower bound?

    That is, inf(A) is unique. Suppose s_1 and s_2 are both inf(A). Since s_1 is a greatest lower bound, s_2 <= s_1.

    But, since s_2 is a greatest lower bound, s_1 <= s_2

    This implies that s_1 = s_2.

    Can't this argument be used for upper bounds...

    Is it because of the notion that if you pick some "upper bound" you can always find something that's higher?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    9
    Quote Originally Posted by seerTneerGevoLI View Post
    How can there be more than 1 upper bound? Can't you use the same argument for showing that there is only 1 greatest lower bound?

    That is, inf(A) is unique. Suppose s_1 and s_2 are both inf(A). Since s_1 is a greatest lower bound, s_2 <= s_1.

    But, since s_2 is a greatest lower bound, s_1 <= s_2
    Let S be a non-empty bounded set of real numbers. By the completness property we have that there exists s_1 which is a least upper bound. Say there is another least upper bound s_2. Now s_2 is an upper bound. Thus, s_1\leq s_2 because s_1 is least. Similarly, s_2\leq s_1. Thus s_1=s_2. Q.E.D.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. proof dealing with supremum and bounds
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: October 20th 2010, 06:09 AM
  2. Supremum
    Posted in the Discrete Math Forum
    Replies: 1
    Last Post: September 14th 2010, 01:02 PM
  3. Supremum
    Posted in the Calculus Forum
    Replies: 2
    Last Post: November 2nd 2008, 11:35 PM
  4. Supremum
    Posted in the Calculus Forum
    Replies: 3
    Last Post: October 22nd 2008, 11:29 AM
  5. Greatest lower bounds and least upper bounds
    Posted in the Calculus Forum
    Replies: 3
    Last Post: March 31st 2008, 06:31 PM

Search Tags


/mathhelpforum @mathhelpforum