# Proof that $\left\lfloor -x\right\rfloor =-\left\lceil x\right\rceil$

I wanted to ask if this kind of reasoning for proving the result in the title could be considered correct:

We know that: $$\left\lceil x\right\rceil =n$$ if and only if $$n-1

Then $$-\left\lceil x\right\rceil =-n$$ if and only if $$-n-1

Then multiplying by $$-1$$ the formula $$-n-1 we get $$n+1>-x\geq n$$ , inverting the sign.

But $$n+1>-x\geq n$$ is equivalent to $$n\leq-x.

We know that $$\left\lfloor x\right\rfloor = n$$ if and only if $$n\leq x.

So from $$n\leq-x we can infer that $$\left\lfloor -x\right\rfloor =$$ $$n=-\left\lceil x\right\rceil$$

• I think you have a couple of errors that cancel out in the end. For instance $f(x)=n\iff something$ is equivalent to $-f(x)=-y\iff something$. So when you change signs the first time to $-\lceil x\rceil=-n$ you should not also change the condition to something NOT equivalent. The condition $-n-1<x\leq -n$ is incorrect. Nov 8, 2021 at 11:31

From the definitions, $$\lfloor x\rfloor=-n\iff -n-1 and $$\lfloor-x\rfloor=n\iff n\le-x

These are equivalent statements.

Alternatively, WLOG the integer part is $$0$$ (because the floor/ceiling commute with addition of an integer), and the fractional part is $$0$$ or $$\in (0,1)$$, say $$0.5$$.

We have $$\lfloor-0\rfloor=-0=-\lceil-0\rceil$$

and

$$\lfloor-0.5\rfloor=-1=-\lceil-0.5\rceil.$$

Your reasoning is quite involved, I think. Try to use the definitions of floor and ceiling directly instead.

By definition, $$\lfloor y \rfloor = k$$ if $$k$$ is the greatest integer such that $$k \leq y$$, and $$\lceil y \rceil = k$$ if $$k$$ is the least integer such that $$y \leq k$$. Moreover, we know that $$x_1 \leq x_2$$ if and only if $$-x_2 < -x_1$$.

So if $$\lfloor -x \rfloor = k$$, then $$k \leq -x$$ and therefore $$x \leq -k$$. Moreover, $$-k$$ must be the least such integer, since $$k$$ is the greatest integer such that $$k \leq -x$$.

Go straight at it, but make careful and clear steps: \begin{align} \lfloor -x\rfloor =n &\iff n\leq -x-n-1 \end{align} and \begin{align} -\lceil x\rceil=n&\iff \lceil x\rceil=-n\\ &\iff -n-1 < x \leq -n \end{align}