Prove that $\lim_{n\to \infty}\int_0^{n^{-\beta}}nf(x)dx=0$, for all $f\in L^q[0,1]$

Prove that $$\lim_{n\to \infty}\int_0^{n^{-\beta}}nf(x)dx=0,\forall f\in L^q[0,1], $$ where $\beta=\frac s{s-1}$, $1<s<q$.

I tried to use Hölder's inequality, but it doesn't work.

Could you please give some hints? Many thanks in advance!

EDIT: I asked this question because I am trying to prove a conlusion in this question. (To make it easier for anyone with the same question to search.)