How to prove the formula for nearest integer for a given real number x?

Question

I'm reading Concrete Mathematics. The problem is from Exercise 3.2 of this book.

The answer is pretty obvious

$$ \lfloor x + 0.5\rfloor, \text{if \{x\} = 0.5 then ceil should be considered}\\ \lceil x - 0.5 \rceil, \text{if \{x\} = 0.5 then floor should be considered} $$ where {x} = fractional part of x

The problem is I'm not able to prove the above equations. How to prove it ?

Answer 1

Since $\lfloor y\rfloor \le y<\lfloor y\rfloor+1$ and there does not exist an integer $n$ with $\lfloor y\rfloor <n<\lfloor y\rfloor+1$, the only candidates for the nearest integer to $y$ are $\lfloor y\rfloor$ and $\lfloor y\rfloor+1$. The distances are $|y-\lfloor y\rfloor|=y-\lfloor y\rfloor =\{y\}$ and $|y-\lfloor y\rfloor-1|=1-y+\lfloor y\rfloor =1-\{y\}$, respectively. To find the nearest, we have to compare $\{y\}$ against $1-\{y\}$ or equivalently, $2\{y\}$ againts $1$. Thus

• If $0\le \{y\}<\frac12$, then the nearest integer is $\lfloor y\rfloor$
• If $\frac12 <\{y\}<1$, then the nearest integer is $\lfloor y\rfloor+1$
• If $\{y\}=\frac12$, then both $\lfloor y\rfloor$ and $\lfloor y\rfloor+1$ are nearest integers.

Remains to show that these piecewise definitions coincide with $\lfloor y+\frac12\rfloor$ (the treatment of $\lceil y-\frac12\rceil$ is similar).

• If $0\le\{y\}<\frac12$ then $y+\frac12<\lfloor y\rfloor +\frac12+\frac12=\lfloor y\rfloor +1$ and hence $\lfloor y+\frac12\rfloor <\lfloor y\rfloor +1$. Together with $\lfloor y+\frac12\rfloor \ge\lfloor y\rfloor$ we conclude that $\lfloor y+\frac12\rfloor = \lfloor y\rfloor$ as desired.
• If $\frac12<\{y\}<1$ then $y+\frac12>\lfloor y\rfloor +\frac12+\frac12=\lfloor y\rfloor+1\in\Bbb Z$ and hence $\lfloor y+\frac12\rfloor \ge\lfloor y\rfloor +1$. Together with $\lfloor y+\frac12\rfloor\le \lfloor y+1\rfloor=\lfloor y\rfloor +1$, we conclude $\lfloor y+\frac12\rfloor =\lfloor y\rfloor +1$.
• If $\{y\}=\frac12$, one readily verifies that $\lfloor y+\frac12\rfloor=\lfloor y\rfloor +1$

In summary, for all $y\in\Bbb R$, $\lfloor y+\frac12\rfloor$ is a nearest integer of $y$.