$X$ is an absolutely continous random variable, with a continous density function, and:
$$X_n= \frac{ \lfloor nX \rfloor}{n}$$
What is the distribution of $X_n$, and what can we say about its convergence?
I tried to calculate the distribution like this:
$$\begin{align*}P(X_n =k)&=P\left(\frac{ \lfloor nX \rfloor}{n}=k\right)=P\left(nk\leq nX <n(k+1)\right)=\int_k^{k+1} f_X\;dx\\&=F_X(k+1)-F_X(k)\end{align*}$$
I am not sure what I did was right, please help!