Does fractional part converge in distribution to a uniform random variable? 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.
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.