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!