Proof ceiling function monotonocity? Is there any indication about the proof of the following statement:
$\forall x,y. x \le y \Rightarrow \left \lceil x \right \rceil \le \left \lceil y \right \rceil$
Yes it's $\le$. Sorry.
Thanks

As noted in the comments, the assertion is false.
What is true is that if $x\le y$, then $\lceil x\rceil\le\lceil y\rceil$. Suppose, on the contrary, that $\lceil x\rceil>\lceil y\rceil$. For convenience let $n=\lceil x\rceil$ and $m=\lceil y\rceil$. Then $n-1<x\le n$ and $m-1\le y<m$, by the definition of the ceiling function. But we also know that $n>m$, where $n$ and $m$ are integers, so $n-1\ge m$, and therefore $x>n-1\ge m>y$, contradicting the hypothesis that $x\le y$.
