Answer
Let $x$ be any non-integral real number, and let $m$ be any integer. Noting that subtraction is simply negative addition and making use Theorem 2.5.1, we have $\lfloor x\rfloor+\lfloor m-x\rfloor=\lfloor x\rfloor + m + \lfloor-x\rfloor$. But from the previous exercise, we know that $\lfloor x\rfloor+\lfloor-x\rfloor=-1$, so we have $\lfloor x\rfloor + m + \lfloor-x\rfloor=m-1$. Hence, by the transitivity of equality, we conclude that $\lfloor x\rfloor+\lfloor m-x\rfloor=m-1$. Since $x$ was an arbitrarily chosen non-integer and $m$ an arbitrary integer, we conclude that the result holds for all non-integers $x$ and integers $m$.
Work Step by Step
Do not be afraid to prove new results by building on previous ones. As shown in the many computer science examples throughout this text, making use of already-proven mathematical results is just like the CS idea of abstraction. Anyone interested in CS applications of discrete mathematics should practice such thinking from both angles.
