Proof ceiling function monotonicity

Question

∀a,b∈R, a ≤ b ⇒ ⌈a⌉ ≤ ⌈b⌉, This statement is true. But if I go about proving it this way- -2.7 ≤ -2, then its ceiling -2 ≤ -2. Is this a valid way to proof this statement or there are some technicalities I have left out??

Answer 1

This is not a proof, it is just an example that seems to support the argument.

In order to build a proof, you should start with the definition of the ceiling function: $\lceil a\rceil$ is the only integer $n\in\mathbb Z$ such that $n-1<a\le n$.

Using this definition, you can prove the statement by considering the different cases and playing with the inequalities.