Let $X_1,X_2,...$ be i.i.d such that $X_i\in\{0,1\}$ with probability $1/2$ each. Let $R_n=\sum_1^n2X_i3^{-i}$.
It is easy to show that $R_n$ converges almost surely, and that the limit takes the values in $[0,1]$ that can be written in ternary (base-$3$) using only $0s$ & $1s$.
I have shown that the limit cannot be a discrete random variable, since the set of possible values is uncountable, and I am also quite sure it is not continuous either, but I cannot think of a straightforward way to show this! How can I show this?
I have only studied basic analysis and fundamentals of Riemann integration so it would be great if any help could not go too complex! Thank you