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Math Help - Suprema proofs

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    Suprema proofs

    1.) Prove that if f and g are bounded above on a nonempty set S, then sup(f+g) is less than or equal to sup f + sup g.

    2.) Give an example of two bounded functions f and g on the interval [0,1] such that sup(f+g) < sup f + sup g.
    Last edited by Slazenger3; January 28th 2010 at 04:56 PM. Reason: adding something else
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by Slazenger3 View Post
    Prove that if f and g are bounded above on a nonempty set S, then sup(f+g) is less than or equal to sup f + sup g.
    f\leqslant \sup\text{ } f and g\leqslant \sup\text{ }g. Therefore, f+g\leqslant \sup\text{ }f+\sup\text{ }g. And thus, \sup\left\{f+g\right\}\leqslant\sup\text{ }f+\sup\text{ }g.

    That was INTENTIONALLY terse. Make it better.
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  3. #3
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by Slazenger3 View Post

    2.) Give an example of two bounded functions f and g on the interval [0,1] such that sup(f+g) < sup f + sup g.
    Think about functions whose maximum points occur at different values. For example, f(x)=\begin{cases} 1 & \mbox{if} \quad x=0 \\ 0 & \mbox{if} \quad x\ne 0\end{cases} and g(x)=\begin{cases} 2 & \mbox{if}\quad x=1 \\ 0 &\mbox{if} \quad x\ne 1\end{cases}.

    Don't use my example. Find one of your own, but use the basic concept behind it.
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