Let $X$ be a continuous random variable with a density function $f(x).$ Let $\{x\}$ denote the fractional part of a real number. I am tryng to prove that $$ \mathbb{P}[\{nX\}\leq z] \rightarrow z, \ \ \forall z \in [0,1] $$
Can anyone help me in this ?
My attempt:
If $g_n(z)$ is the density of $\{nX\}$, I got $g_n(z)=\sum_{m \in \mathbb{Z}}\frac{1}{n}f(\frac{m}{n}+\frac{z}{n})$. Can I justify from here ?
One can assume nice properties on $f$ if needed but I want to keep those assumptions as minimal as possible.
The obvious counterexample is the constant case $f(x) = \delta_{1/2}$ (for the Dirac $\delta$). More generally, I doubt this is true of pathological continuous densities constructed, say, with peaks at all dyadic rational numbers.
If you specify extra smoothness then it may be true.