Suprema proofs

**Slazenger3** · January 28th 2010, 04:53 PM

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.

**Drexel28** · January 28th 2010, 04:57 PM

Originally Posted by Slazenger3

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.

**Drexel28** · January 28th 2010, 05:00 PM

Originally Posted by Slazenger3

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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