\(\displaystyle x\in\mathbb{R}, n\in\mathbb{Z}\)

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

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: \(\displaystyle x\notin\mathbb{Z}\)

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

\(\displaystyle k+1+n=k+1+n\)