I'm trying to prove the next:
If $X$ is cád (right continuous) and adapted process, then $\displaystyle\lim_{n\rightarrow\infty}X_{Z_{n}}=X_{T}$ and $X_{T}$ is random variable.
Here $T$ is a stopping time such that $\displaystyle\lim_{n\rightarrow\infty}Z_{n}(\omega)=T(\omega),$ where $$Z_{n}(\omega)=\left\{ \begin{array}{lcc} \frac{X}{2^{n}} & if & \frac{k-1}{2^{n}}\leq T(\omega)<\frac{k}{2^{n}},\space n=1,2,\ldots \\ \\ \infty & if & T(\omega)=\infty. \end{array} \right.$$
I cannot see how the right continuos of the process works on this sequence to prove the desire limit. Also, $X_{T}$ will be random variable because is limit of random variables, isn't?
Any kind of help is thanked in advanced.