Let $\displaystyle b>0$.
Compute the next limit:
$\displaystyle lim_{n\to \infty}\frac{n}{b}[\frac{n}{b}]$
My starting:
$\displaystyle 1-\frac{n}{b}<\frac{n}{b}[\frac{n}{b}]<1$
And I stuck here...
(By the way I think the limit is 0)
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Let $\displaystyle b>0$.
Compute the next limit:
$\displaystyle lim_{n\to \infty}\frac{n}{b}[\frac{n}{b}]$
My starting:
$\displaystyle 1-\frac{n}{b}<\frac{n}{b}[\frac{n}{b}]<1$
And I stuck here...
(By the way I think the limit is 0)
We have $\displaystyle \left[\frac{n}{b}\right]\leq\frac{n}{b}<\left[\frac{n}{b}\right]+1$
Then $\displaystyle \left[\frac{n}{b}\right]>\frac{n}{b}-1$
Multiply both members by $\displaystyle \frac{n}{b}$:
$\displaystyle \frac{n}{b}\left[\frac{n}{b}\right]>\frac{n}{b}\left(\frac{n}{b}-1\right)$
But $\displaystyle \displaystyle\lim_{n\to\infty}\frac{n}{b}\left(\fr ac{n}{b}-1\right)=\infty$
and applying the limit in the inequality we have that $\displaystyle \displaystyle\lim_{n\to\infty}\frac{n}{b}\left[\frac{n}{b}\right]=\infty$
