# Prove that for every $p>0$, $\lim_{n\rightarrow∞}\int_n^{n+p}{\sin x\over x} = 0$

Got stuck with this question:

Prove that for every $p>0$, $\displaystyle \lim \limits_{n\rightarrow∞}\int_n^{n+p}{\sin (x)\over x} = 0$.

Thanks in advance for any help!

Note that for $n>0$, $$\left|\int_n^{n+p} \frac{\sin(x)}{x}\,\mathrm{d}x\right|\leq \int_n^{n+p}\frac{1}{x}\,\mathrm{d}x = \ln(n+p)-\ln(n) = \ln\left(1+\frac{p}{n}\right).$$