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.
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.
$\displaystyle f\leqslant \sup\text{ } f$ and $\displaystyle g\leqslant \sup\text{ }g$. Therefore, $\displaystyle f+g\leqslant \sup\text{ }f+\sup\text{ }g$. And thus, $\displaystyle \sup\left\{f+g\right\}\leqslant\sup\text{ }f+\sup\text{ }g$.
That was INTENTIONALLY terse. Make it better.
Think about functions whose maximum points occur at different values. For example, $\displaystyle f(x)=\begin{cases} 1 & \mbox{if} \quad x=0 \\ 0 & \mbox{if} \quad x\ne 0\end{cases}$ and $\displaystyle g(x)=\begin{cases} 2 & \mbox{if}\quad x=1 \\ 0 &\mbox{if} \quad x\ne 1\end{cases}$.
Don't use my example. Find one of your own, but use the basic concept behind it.