Let $X_1,...X_n$ be a sequence of random variables such that $X_n=1 $ or $0$ and $P(X_1=1) \geq \alpha$ and $P(X_n=1|X_1,...X_{n-1}) \geq \alpha$ for $n=2,3,...$ where $\alpha >0$
I need to show $P(X_n=1$ infinitely often )$ =1$
Using Borel-Cantelli lemma,
I can show that $\sum P(X_n=1) =\infty $ because
$\sum P(X_n=1) = P(X_1=1) + P(X_2 =1| X1) + P(X_3|X_1,X2) + .... > \alpha + \alpha + .... =\infty $
But to apply the Borel-Cantelli lemma, the sequence should be independent . But in this case it is not.
Can anyone help me to figure out how to find an independent sub sequence ? Also is there any better approach than Borel-Cantelli lemma for this question?