Inequality with the difference of ceil functions

Question

I am interested in the validation of the inequality $$ \lceil x \rceil - \lceil y \rceil \leqslant \lceil x-y \rceil $$ where $x, y$ are assumed to be positive.

Can someone help me to proof it?

Answer 1

$x\le\lceil x \rceil< x+1\Rightarrow \lceil x \rceil<x+1$

$y\le\lceil y \rceil< y+1 \Rightarrow -\lceil y \rceil \le -y$

$\Rightarrow \lceil x\rceil-\lceil y\rceil < x-y+1 \le \lceil x-y\rceil+1$

How $\lceil x\rceil-\lceil y\rceil$ and $\lceil x- y\rceil+1$ are integers, then $\lceil x\rceil-\lceil y\rceil < \lceil x-y\rceil+1 \Rightarrow \lceil x\rceil-\lceil y\rceil \le \lceil x-y\rceil$.

Answer 2

Note $y\le \lceil y\rceil$ and $x-y\le\lceil x-y\rceil$, so (by adding those two inequalities):

$$x=y+(x-y)\le\lceil y\rceil+\lceil x-y\rceil$$

Thus, $\lceil y\rceil+\lceil x-y\rceil$ is one integer greater than or equal $x$. $\lceil x\rceil$ is the smallest such integer. Therefore $\lceil x\rceil\le\lceil y\rceil+\lceil x-y\rceil$, which is equivalent to what you have to prove.