<s>[INST] What is the relation between $\lfloor x+y \rfloor$ and $\lfloor x \rfloor + \lfloor y\rfloor$ for any reals $x,y$ What is the relation between $\lfloor x+y \rfloor$ and $\lfloor x \rfloor + \lfloor y\rfloor$ for any reals $x,y$?    
My effort: 
We have $x+y-1\lt \lfloor x+y\rfloor \leq x+y$. What can I do after this? Please help
 [/INST] As by definition, $\,\lfloor x\rfloor\le x <\lfloor x\rfloor+1$, we deduce
$$\lfloor x\rfloor+\lfloor y\rfloor\le x+y<\lfloor x\rfloor+\lfloor y\rfloor+2, $$
whence
$$\lfloor x\rfloor+\lfloor y\rfloor\le \lfloor x+y\rfloor\le\lfloor x\rfloor+\lfloor y\rfloor+1.$$
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