Question
Let $p>1$, $I_n = (-1/n,1/n)$ and $\gamma \le (p-1)/p$. Assume that $f \in L^p(I_2,dx)$. Calculate $\lim_{n\rightarrow \infty}n^\gamma \int_{I_n}|f(x)|dx$

I could show that
$n^\gamma \int_{I_n}|f(x)|dx \le n^\gamma \|f \|_p (2/n)^{(p-1)/p}$
but I cannot find where it converges in limit. I would assume it will converge to 0.