# [SOLVED] Ceiling Functions

#### dwsmith

MHF Hall of Honor
$$\displaystyle \left \lceil x+n \right \rceil=\left \lceil x \right \rceil+n$$

$$\displaystyle x\in\mathbb{R}, n\in\mathbb{Z}$$

$$\displaystyle x=k+x', 0\leq x'<1$$

Case 1: $$\displaystyle x\in\mathbb{Z}$$
$$\displaystyle \left \lceil x+n \right \rceil=\left \lceil x \right \rceil+n\rightarrow \left \lceil k+x'+n \right \rceil=\left \lceil k+x' \right \rceil+n$$
$$\displaystyle k+n=k+n$$

Case 2: $$\displaystyle x\notin\mathbb{Z}$$
$$\displaystyle \left \lceil x+n \right \rceil=\left \lceil x \right \rceil+n\rightarrow \left \lceil k+x'+n \right \rceil=\left \lceil k+x' \right \rceil+n$$
$$\displaystyle k+1+n=k+1+n$$

#### chiph588@

MHF Hall of Honor
If $$\displaystyle x\in\mathbb{Z}$$ then $$\displaystyle x+n\in\mathbb{Z} \implies \left \lceil x+n \right \rceil = x+n = \left \lceil x \right \rceil + n$$

If $$\displaystyle x\not\in\mathbb{Z}$$ then $$\displaystyle x+n\not\in\mathbb{Z} \implies \left \lceil x+n \right \rceil = \left \lfloor x+n \right \rfloor + 1 = x+n+\{x+n\}+1 = x+n+\{x\}+1 =$$ $$\displaystyle x+\{x\}+1+n = \left \lceil x \right \rceil +n$$

#### dwsmith

MHF Hall of Honor
If $$\displaystyle x\in\mathbb{Z}$$ then $$\displaystyle x+n\in\mathbb{Z} \implies \left \lceil x+n \right \rceil = x+n = \left \lceil x \right \rceil + n$$

If $$\displaystyle x\not\in\mathbb{Z}$$ then $$\displaystyle x+n\not\in\mathbb{Z} \implies \left \lceil x+n \right \rceil = \left \lfloor x+n \right \rfloor + 1 = x+n+\{x+n\}+1 = x+n+\{x\}+1 =$$ $$\displaystyle x+\{x\}+1+n = \left \lceil x \right \rceil +n$$
I was using some definition in my book so that is where I came up with x=k+x', so the question then becomes is what I have original correct? If not, why?

#### chiph588@

MHF Hall of Honor
I was using some definition in my book so that is where I came up with x=k+x', so the question then becomes is what I have original correct? If not, why?
To be honest, I was having trouble following your work clearly. Care to elaborate a bit? (Itwasntme)

#### dwsmith

MHF Hall of Honor
To be honest, I was having trouble following your work clearly. Care to elaborate a bit? (Itwasntme)
$$\displaystyle \forall x\in\mathbb{R}$$ can be written as $$\displaystyle x=k+x'$$, where $$\displaystyle k=\left \lfloor x \right \rfloor$$ and $$\displaystyle 0\leq x'<1$$

That is from the book.

So then I did the substitution incorporating that definition.

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#### chiph588@

MHF Hall of Honor
$$\displaystyle \forall x\in\mathbb{R}$$ can be written as $$\displaystyle x=k+x'$$, where $$\displaystyle k=\left \lfloor x \right \rfloor$$ and $$\displaystyle 0\leq x'<1$$

That is from the book.

So then I did the substitution incorporating that definition.
I got that far, but what is your reasoning after that?

#### dwsmith

MHF Hall of Honor
I got that far, but what is your reasoning after that?
The equality then holds since the LHS=RHS.

#### chiph588@

MHF Hall of Honor
Ok I see it now, but for this to be a formal proof, you'll need more of an explanation of how you got to the last line of each case.

dwsmith

#### dwsmith

MHF Hall of Honor
Ok I see it now, but for this to be a formal proof, you'll need more of an explanation of how you got to the last line of each case.
Boo, I hate writing words.