1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Inequality of Supremums

  1. Sep 17, 2005 #1

    For every non empty set E of real numbers that is bounded above there exists a unique real number sup(E) such that

    1. sup(E) is an upper bound for E.

    2. if y is an upper bound for E then y [itex]\geq[/itex] sup(E).


    [itex]sup(A\cap B)\leq sup(A)[/itex]

    I can show a special case of this,

    if [itex]A\cap B=\emptyset [/itex], then [itex]sup(A\cap B)\leq sup(A)[/itex].

    Nothing is less than something, right?

    Now here's my problem...
    Beyond the trivial case, all I have been able to do is draw pictures of sets on a number line. The pictures make the inequality really obvious, but I don't think that pictorial intuition counts as a real proof.

    Could anyone give me a pointer on how to set up a real proof?

    Last edited: Sep 17, 2005
  2. jcsd
  3. Sep 17, 2005 #2
    Ok break that inequality into two parts
    first prove the equal to part. How would you do that . Assume the su pof A and B are something and that they are equal. That way you can prove the equality part
    Then assume that the sup of A is greater than the Sup B. Now what is sup (a inter B) ? What is it less than?
  4. Sep 17, 2005 #3
    Ok, this is what I have for the equality part if I'm understanding you right.

    let [tex]sup(A)=sup(B)[/tex]

    then [tex]sup(A\cap B)=sup(A)[/tex]

    And the inequality would look like this?

    let [tex]sup(B)\leq sup(A)[/tex]

    then [tex]sup(A\cap B)\leq sup(A)[/tex]

    Is that right?
    It seems too simple...
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?

Similar Discussions: Inequality of Supremums
  1. Inequality question (Replies: 2)

  2. Inequality problem (Replies: 5)

  3. Inequality question (Replies: 8)

  4. Proving an Inequality (Replies: 3)