I'm working on this exercise at the moment:
Let $ X: \Omega \rightarrow (0,1) $ be a uniformly distributed random variable. Define $X_n : \Omega \rightarrow (0,1)$ as $$ X_n =\begin{cases} 0 \text{ if } \lfloor2^n X\rfloor \text{ is even}\\ 1 \text{ if } \lfloor2^n X\rfloor \text{ is odd}. \end{cases}$$ Show that $X_1,X_2,\dots$ is an independent sequence of Bernoulli$(1/2)$-distributed random variables.
So far I have shown that $X_1,X_2,\dots $ are Bernoulli$(1/2)$-distributed, but I am lacking an approach for showing the independence.