# sup and inf

• Mar 29th 2011, 02:59 PM
Connected
sup and inf
Let $\displaystyle f,g:A\to\mathbb R$ so that $\displaystyle f\le g,$ then prove that if $\displaystyle g$ is bounded above, so is $\displaystyle f$ and $\displaystyle \inf f\le \sup g.$

First part is easy, now since $\displaystyle \inf f\le f\le g,$ we have that $\displaystyle \inf f\le g\implies \inf f\le\sup g.$ Is this correct?

Now suppose $\displaystyle f$ is bounded below, we have $\displaystyle \inf f\le \inf g.$

If $\displaystyle f$ is bounded below so does $\displaystyle g$ then $\displaystyle \inf f\le f\le g,$ where $\displaystyle \inf f\le g\implies \inf f\le \inf g.$ Is it correct?

Thanks!
• Mar 29th 2011, 08:06 PM
bkarpuz
Quote:

Originally Posted by Connected
Let $\displaystyle f,g:A\to\mathbb R$ so that $\displaystyle f\le g,$ then prove that if $\displaystyle g$ is bounded above, so is $\displaystyle f$ and $\displaystyle \inf f\le \sup g.$

First part is easy, now since $\displaystyle \inf f\le f\le g,$ we have that $\displaystyle \inf f\le g\implies \inf f\le\sup g.$ Is this correct?

Correct
Quote:

Originally Posted by Connected
Now suppose $\displaystyle f$ is bounded below, we have $\displaystyle \inf f\le \inf g.$

If $\displaystyle f$ is bounded below so does $\displaystyle g$ then $\displaystyle \inf f\le f\le g,$ where $\displaystyle \inf f\le g\implies \inf f\le \inf g.$ Is it correct?

Correct

In such questions, I can suggest you to always mention the domains of the variables, i.e.,
Assume that $\displaystyle f(t)\leq g(t)$ for all $\displaystyle t\in A$, then
we have
$\displaystyle \inf_{t\in A}\{f(t)\}\leq f(t)\leq g(t)$ for all $\displaystyle t\in A$,
which implies that (dont forget that $\displaystyle \inf_{t\in A}\{f(t)\}$ is constant and is independent of $\displaystyle t$)
$\displaystyle g$ is bounded below by a constant $\displaystyle \inf_{t\in A}\{f(t)\}$, and thus,
$\displaystyle \inf_{t\in A}\{f(t)\}\leq\inf_{t\in A}\{g(t)\}\leq g(t)$ for all $\displaystyle t\in A$.
• Mar 30th 2011, 03:02 AM
Connected
Yes of course, it's just to save writing, that's all!