<s>[INST] If $|a-b| \leq C$ and $|x-y|\leq D$, is there a way to bound $\left|\frac{a}{x} - \frac{b}{y}\right|$? If $|a-b| \leq C$ and $|x-y|\leq D$, is there a nice way to bound $\left|\frac{a}{x} - \frac{b}{y}\right|$?
I made an attempt using Triangle inequality but to no avail.
\begin{align*}\left|\frac{a}{x} - \frac{b}{y}\right| &= \left|\frac{a}{x} - \frac{b}{x} + \frac{b}{x} - \frac{b}{y}\right| \\
&\leq \frac{|a-b|}{|x|} + |b| \left|\frac{y - x}{xy}\right| \\
&\leq \frac{C}{|x|} + \frac{|b|D}{|xy|}
\end{align*}
Is there a nicer one?
 [/INST] Think about this intuitively.  Consider $|\frac{a}{x} - \frac{b}{y}|$.  Let's just keep $a$ and $b$ fixed for now.  Consider $|x - y| \leq D$, which basically means $-D \leq x-y \leq D$.  If we choose $x$ and $y$ from $(-\frac{D}{2}, \frac{D}{2})$, then we will always have $x-y$ in $(-D,D)$ (why?).
Okay, but let's fix $y$ for a moment.  We can choose $x$ as close to $0$ as we want.  That will make $|\frac{a}{x} - \frac{b}{y}|$ grow unboundedly toward $\infty$ as $x$ gets closer and closer to $0$.  So, we've seen that under all of your assumptions, we can find an example where $|\frac{a}{x} - \frac{b}{y}|$ is actually an unbounded expression.
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