I want to prove that if $a$ and $b$ are positive integers, the following equality is true:
$$\left\lceil \frac{a}{b}\right\rceil = \left\lfloor\frac{a+b-1}{b}\right\rfloor$$
What have I thought about?
Then $a$ = $qb + r$, where $q$ is the quotient and $r$ the remainder
\begin{align} b-1 &\ge r \ge 0 \\ qb + (b-1) &\ge qb+r \ge qb \\ qb + (b-1) &\ge a\ge qb \\ \left\lfloor\frac{a}{b} \right\rfloor +1-\frac1{b} &\ge \frac{a}{b} \ge \left\lfloor\frac{a}{b}\right\rfloor \\ \left\lfloor \frac{a+b}{b} \right\rfloor -\frac1{b} &\ge \frac{a}{b} \ge \left\lfloor\frac{a}{b}\right\rfloor \end{align}
