Question 11
For a real number $x$ let $[x]$ denote the greatest integer not greater than $x$.
a. Prove that for all positive integers $a$, $b$ and $c$ we have
$$
\left\lfloor \frac{\left\lfloor \frac{c}{a} \right\rfloor}{b} \right\rfloor \le \left\lfloor \frac{c}{ab} \right\rfloor.
$$
b. Find an example showing that the above equality does not hold for all positive real numbers $a$, $b$ and $c$.
a. Prove that for all positive integers $a$, $b$ and $c$ we have
$$
\left\lfloor \frac{\left\lfloor \frac{c}{a} \right\rfloor}{b} \right\rfloor \le \left\lfloor \frac{c}{ab} \right\rfloor.
$$
b. Find an example showing that the above equality does not hold for all positive real numbers $a$, $b$ and $c$.