How to prove that the $\lceil xy \rceil \le \lceil x \rceil \lceil y \rceil$ for positive real numbers $x, y$?

Question

I know intuitively that this is true, but whenever I try and break apart that intuition to see where it's coming from, I essentially end up re-writing the assumption I'm trying to prove.

I've tried playing around with the definition that we use for the ceiling function (given $x \in \mathbb{R}$, there exists an integer $n$ such that $n-1 < x \leq n < x+1$), but that's been fruitless so far, as well.

Found stuff on Wikipedia/Wolfram pertaining to sums and floor/ceiling functions, but nothing pertaining to products and floor/ceiling functions.

---

Proof attempt

For any real numbers $x$ and $y$, we can let them be equal to the sum of integers $k_1$, $k_2$ and remaining decimal values $d_1$, $d_2$.

The proof will hinge on the fact that the decimal values $d_1$, $d_2$ will always fall in the interval $[0,1)$.

You then look at the cases exhausting the possible combinations of values for $d_1$ and $d_2$, most importantly both $= 0$ or both $> 0$ (the boundary cases). In each of these cases, plug in $(k_1 + d_1)$ for $x$ and $(k_2 + d_2)$ for $y$ and expand LHS and RHS of your inequalities until they have reached the necessary form.

Bear in mind that in our integer + decimal form, $\lceil k + d \rceil = k$ if $d=0$ and $= k+1$ if $d>0$.

Answer 1

The ceiling of a number is the least integer greater than or equal to it. In particular, this means that if $n \geq x$ for $n$ an integer and $x$ a real number then $n$ is greater than or equal to the ceiling of $x$. Thus to prove your inequality, all you have to show is that $\lceil x \rceil * \lceil y \rceil \geq xy$.

Answer 2

It is obvious that ceiling of a real number is at least as much as that number, therefore, $Ceiling(x)Ceiling(y)\geq xy$. As $Ceiling(xy)$ is defined to be such minimum integer, we have $Ceiling(x)Ceiling(y)\geq Ceiling(xy)$.

Answer 3

If $x \geq 0$ and $y \geq 0$, then $\lceil x \rceil \lceil y \rceil \geq xy$.

Example: $x=1/2$ and $y=-1$ $$\lceil xy\rceil=\left\lceil \frac{1}{2}\cdot(-1)\right\rceil=\left\lceil -\frac{1}{2}\right\rceil=0\,,$$ whereas $$\lceil x\rceil\lceil y\rceil=\left\lceil \frac{1}{2}\right\rceil\left\lceil -1\right\rceil=(1)(-1)=-1\,.$$