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Exercises
1
No. The statement holds if and only if $x$ is not integer. For example $\lceil 1 \rceil = 1 \neq 1 + \lfloor 1 \rfloor = 2$.
Let $x = z + y$ where $z$ is integer and $0 < y < 1$. Then $\lceil x \rceil = z + 1 = \lfloor x \rfloor + 1$.