The goal is to prove that $\lceil x+m\rceil=\lceil x\rceil +m$, where $x$ is a real number and $m$ is an integer. The book outlines the following proof:
Write $x=n-\epsilon$, where $n$ is an integer and $0\leq\epsilon<1$; thus, $\lceil x\rceil=n$. Then $\lceil x+m\rceil=\lceil n-\epsilon+m\rceil=n+m=\lceil x\rceil+m$.
If we "read between the lines," it really seems like the following is being communicated:
\begin{align} \lceil x+m\rceil&=\lceil n-\epsilon+m\rceil\\ &= \lceil-\epsilon\rceil+(n+m)\tag{circular reasoning?}\\ &= 0+(n+m)\tag{since $\epsilon\in[0,1)$}\\ &= \lceil x\rceil+m \end{align}
Is the book's proof fine and I'm just not seeing something clearly or is there a subtle error somewhere (if so, what could be done to fix the proof?)?
As is pointed out in the comments, there is no circular reasoning here. To clear things up substitute $u=n+m$ (which will be an integer) and then use the theorem you already used and accepted, that $\lceil n-\epsilon \rceil = n$ where $n$ is any integer (and replace $n$ by $u$)