Problem: For all $x \in R$ $\lfloor x \rfloor \not= \lceil x \rceil$
I believe I need to utilize the definitions of both, whereas;
floor of x equals unique integer n s.t. n less than or equal to x less than 1;
ceiling of x equals unique integer n s.t. n-1 less than x less than or equal to n
I can quickly find an example to show there exists a real number (such as 1.2) that shows the floor and ceiling are not equal. However, I realize this is not a proof.
I do not believe (or at least found) a counter example.
I hope I have provided sufficient information. Any help would be greatly appreciated.