Answer
$\lceil {x+m}\rceil=\lceil {x}\rceil + m$
Work Step by Step
Given: x is a real number and m is an integer
To prove:$\lceil {x+m}\rceil=\lceil {x}\rceil + m$
Proof:
FIRST PART- Let x be an integer. Since m is an integer, then x+m is also
an integer.
The ceiling function of an integer is the integer itself
$$\lceil {x}\rceil=x$$ $$\lceil {x+m}\rceil=x + m$$
Thus we have then derived:
$$\lceil {x+m}\rceil=x + m=\lceil {x}\rceil+m$$
SECOND PART- Let x is not an integer. Any real number x is lies between
two consecutive integers (not inclusive, since x is not an integer)
$$\exists n \in Z:n
