$\displaystyle \left \lceil x+n \right \rceil=\left \lceil x \right \rceil+n$

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

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

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: $\displaystyle x\notin\mathbb{Z}$

$\displaystyle \left \lceil x+n \right \rceil=\left \lceil x \right \rceil+n\rightarrow \left \lceil k+x'+n \right \rceil=\left \lceil k+x' \right \rceil+n$

$\displaystyle k+1+n=k+1+n$