I was trying to solve the following problem:
Suppose $h\colon [0,1]\to \mathbb{R}$ is a continuous function such that $\int_0^1 h(x)dx=0$. For $x\ge 0$, let $\{ x \}$ represent the fractional part of $x$: $\{ x \}=x-p$, where $p$ is a positive integer such that $p\le x<p+1$. Show that
a) For any $0\le a \le b\le 1$, we have $\lim_{n\to\infty}\int_a^b h(\{ nx \})dx=0$.
b) For any $g\in L^1[0,1]$, we have $\lim_{n\to\infty}\int_0^1 h(\{ nx \})g(x)dx=0$.
My initial idea was to use the Monotone Convergence Theorem by defining $h_n:=h(\{ nx \})$ on $[a,b] \subseteq [0,1]$, but I am struggling to justify why this would converge to zero (not really sure how to work with the fractional part here). Any help would be appreciated.
