Disprove the following statement:
For all real numbers $x$ and $y$, if $x + \lfloor x \rfloor = y + \lfloor y \rfloor$ then $x = y$.
Aka: Prove the negation:
There are real numbers $x$ and $y$, that $x + \lfloor x \rfloor = y + \lfloor y \rfloor$ and $x \neq y$.
I have tried plugging in many real number combinations to disprove it, and have tried a few properties to try and prove this but I am completely stuck!
Any hints or guidance on how to approach this question would be helpful in the least.
Best Answer
The claim you're trying to disprove looks true to me. (The function $x\mapsto x+\lfloor x\rfloor$ is strictly increasing and therefore injective). So you shouldn't be able to disprove it.