limit problem...

We have $\left[\frac{n}{b}\right]\leq\frac{n}{b}<\left[\frac{n}{b}\right]+1$

Then $\left[\frac{n}{b}\right]>\frac{n}{b}-1$

Multiply both members by $\frac{n}{b}$ :

$\frac{n}{b}\left[\frac{n}{b}\right]>\frac{n}{b}\left(\frac{n}{b}-1\right)$

But $\displaystyle\lim_{n\to\infty}\frac{n}{b}\left(\fr ac{n}{b}-1\right)=\infty$

and applying the limit in the inequality we have that $\displaystyle\lim_{n\to\infty}\frac{n}{b}\left[\frac{n}{b}\right]=\infty$