1. Limited time only! Sign up for a free 30min personal 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!

Suprema Problem

  1. Jun 19, 2013 #1
    Let X and Y be nonempty sets and let h: X x Y→ℝ
    Define f: X →ℝ and g: Y→ℝ by the following:

    f(x)=sup{h(x,y): y in Y} and g(y) = inf{h(x,y): x in X}

    Prove that sup{g(y): y in Y} ≤ inf{f(x): x in X}

    Attempt at solution:
    Pick y' in Y. Then g(y')≤h(x,y') for all x in X. Hence, there exist some x' such that g(y')≤h(x',y').
    Then, h(x',y')≤ sup{h(x',y): y in Y} = f(x')..... Not too sure where to go from here... A small hint would be great!

  2. jcsd
  3. Jun 19, 2013 #2
    Wait... I think I got it..

    Pick any y' and x'. Then, g(y') ≤ h(x,y') for all x; therefore g(y')≤h(x',y'). Now, h(x',y')≤sup{h(x',y) : y in Y}=f(x'). This established the following inequality:


    Because x' and y' were arbitrary, we conclude g(y)≤f(x) for all x,y. Thus g(y) is a lower bound for the set {f(x): x in X} for all y; it follows that g(y)≤inf{f(x):x in X} for all y. Then, since g(y)≤inf{f(x):x in X}, inf{f(x):x in X} is an upper bound for the set {g(y): y in Y} ....... the rest is history! :)
  4. Jun 19, 2013 #3


    User Avatar
    Science Advisor
    Homework Helper

    Sounds ok to me.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted

Similar Threads - Suprema Problem Date
Integration problem using u substitution Monday at 1:02 PM
Maximum/minimum problem Saturday at 4:00 AM
Minimum/Maximum problem Thursday at 1:55 PM
Equivalence of maps on l-infinity (involves limits, suprema and sums) Dec 12, 2012
Suprema proof: quick check please Sep 3, 2011