Let $\{X_n : n \ge 1 \}$ and $\{Y_n : n \ge 1 \}$ be two i.i.d Bernoulli random variables with parameters $p_1, p_2$, respectively (where $X_n \perp Y_n$ $\forall n$)
Define $Z_n = X_nY_n$ for $n \ge 1$. I've been able to prove that the distribution of $Z_n$ is still Bernoulli with parameter $p_1p_2$; however, I'm unsure if the sequence $\{Z_n : n \ge 1 \}$ maintains the independence property. I've tried to prove or disprove $P(Z_{n_1} = a \cap Z_{n_2} = b) = P(Z_{n_1} = a)P(Z_{n_2} = b)$ for $n_1 \neq n_2$, but I haven't been able to make a lot of progress.