Proving that $ \lfloor \lceil x \rceil \rfloor = \lceil x \rceil $ for all real numbers $ x $.

Question

Let $ \mathsf{LHS} = \lfloor \lceil x \rceil \rfloor $ and $ \mathsf{RHS} = \lceil x \rceil $.

Let us call $ n = \lceil x \rceil $.

Case 1: $ n $ is even, i.e., there exists a $ k \in \mathbb{Z} $ such that $ n = 2 k $.

I’m not sure how to go on or if I’m setting this up right.

Answer 1

$\lceil x\rceil$ is an integer, and if $n$ is an integer then $\lfloor n\rfloor=n$.

Answer 2

First notice that $ \left \lfloor{n}\right \rfloor = n$ for all integers $ n$. And $ \lceil x \rceil $ is an integer