$ \frac{1}{t_n} \int_0^{t_n} e^{2 \pi i h f(x)} \, dx \to 0$ if $ \frac{1}{n} \int_0^n e^{2 \pi i h f(x)} \, dx \to 0 $

Question

Suppose that $f$ is a measurable functions on the real line and $h \in \mathbb Z \setminus \{ 0 \}$ such that $$ \frac{1}{n} \int_0^n e^{2 \pi i h f(x)} \, dx \to 0 $$ as $n \in \mathbb N$ tends to infinity. I want to show that if $(t_n)_n$ is an arbitrary sequence tending to infinity then also $$ \frac{1}{t_n} \int_0^{t_n} e^{2 \pi i h f(x)} \, dx \to 0. $$ I read this in a paper without proof. Right now I don't see how to approach this problem. Any help is welcome.