For a random variable $X$ with a density function $f(x),$ I want to prove that the following holds: $$ \lim_{n \rightarrow \infty}I(\lfloor nX\rfloor;\{nX\})=0 $$
where $\lfloor x \rfloor, \{x\}$ denote the integer and fractional parts of a real number $x$.
My attempt: I was able to prove that $\{nX\}$ converges to uniform random variable in distribution. But I couldn't use this information to prove the desired result.
Can anyone help me with this ? One can assume additional properties on $f(x)$ if needed.