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It has been given that $x$ is a real number and $m$ is an integer.

if $x$ is a real number then the number can be represented as $y.z$ where $y$ is the integral part and $z$ is the fractional part.

$\lceil \ x+m \ \rceil = \lceil \ (y+m).z \ \rceil = y+m+1$

and , $\lceil \ x \ \rceil +m  = \lceil \ y.z \ \rceil+m = y+1+m$.


Thus both give the same result.

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