Difference of 2 floors vs floor of difference

Good day, we are currently covering basic principles for algorithm optimisation and we were tasked with explaining the following problem.

Assume $$x,y \in \Bbb R$$. How much can the following two integers differ:

$$\lfloor x \rfloor - \lfloor y \rfloor$$ and $$\lfloor x-y \rfloor$$

These can clearly be equal but I am not quite sure how to explain or approach the explanation of how they differ.

They can differ by at most the difference between the integer parts of $$x$$ and $$y$$ but I do not think that this is enough.

Write $$x=n+a$$ and $$y=m+b$$ with $$n,m\in\mathbb Z$$ and $$0\le a,b<1$$. $$\lfloor x\rfloor-\lfloor y\rfloor$$ is obviously $$n-m$$, while $$\lfloor x-y\rfloor=\lfloor n-m+(a-b)\rfloor=n-m+\lfloor a-b\rfloor$$ From the bounds on $$a,b$$ we see that $$-1, so $$\lfloor a-b\rfloor$$ is 0 if $$0\le a-b<1$$ and $$-1$$ otherwise.
Therefore $$\lfloor x\rfloor-\lfloor y\rfloor$$ is either equal to or one greater than $$\lfloor x-y\rfloor$$.