Why Does Boundedness of a Function Affect Its Oscillation?

[Arkuski]

Suppose that $f$ is bounded by $M$. Prove that $ω(f^2,[a,b])≤2Mω(f,[a,b])$.

I can show that $ω(f,[a,b])≤2M$ and that $ω(f^2,[a,b])≤M^2$ but this procedure is getting me nowhere. I also have a similar problem that likely calls for the same approach:

Suppose that $f$ is bounded below by $m$ and that $m$ is a positive number. Prove that $ω(1/f,[a,b])≤ω(f,[a,b])/m^2$.

This one I think I have right but my instructor is telling me that it's wrong. Since all values are positive, by the nature of $\frac{1}{x}$, $\displaystyle\sup f = \frac{1}{\displaystyle\inf f}$ and $\displaystyle\inf f = \frac{1}{\displaystyle\sup f}$. We can now analyze the oscillation as follows:

$ω(1/f,[a,b])=\frac{1}{\displaystyle\inf f}-\frac{1}{\displaystyle\sup f}=\frac{ω(f,[a,b])}{(\displaystyle\inf f)(\displaystyle\sup f)}≤\frac{ω(f,[a,b])}{m^2}$

 

[haruspex]

Suppose that $f$ is bounded by $M$. Prove that $ω(f^2,[a,b])≤2Mω(f,[a,b])$.

It's probably not the most elegant, but you could try breaking it into separate cases according to the signs of sup f and inf f.

$\displaystyle\sup f = \frac{1}{\displaystyle\inf f}$

I think you mean $\displaystyle\sup \frac{1}{f} = \frac{1}{\displaystyle\inf f}$ etc. Other than that, your proof looks fine.