convert ceil to floor Mathematically, why is this true?
$$\left\lceil\frac{a}{b}\right\rceil= \left\lfloor\frac{a+b-1}{b}\right\rfloor$$
Assume $a$ and $b$ are positive integers.
Is this also true if $a$ and $b$ are real numbers?

Suppose, $b\nmid a$. Let $\lfloor\frac{a+b-1}{b}\rfloor=n \implies nb+r=a+b-1$ where $0\leq r<b$. So $\frac{a}{b}+1=n+\frac{r+1}{b}$. So $\lfloor\frac{a}{b}+1\rfloor=\lceil\frac{a}{b}\rceil=n$.
This is not true in general as shown by Lyj.
