Show convergence a.s. when sum of $P(A_n)$ is $\infty$ and the sequence is not independent

Question

There is a sequence of random variables defined by $$Y_n = \Big(\Big|{1-\frac\Theta \pi}\Big|\Big)^n$$ where $\Theta\sim\mathrm{unif}[0,2\pi].$
I have shown that the sequence converges to $0$ in probability. Applying Borel Cantelli Lemma by taking $A_n = \{|Y_n|>0\}$ and then evaluating $\sum_{n=1}^\infty P(A_n)$, I get it to be $\infty$. Since the random variables are not independent, I can't conclude anything from this result. How do I prove/disprove convergence almost surely?

Answer 1

$Z=(1-\Theta/\pi)$ is almost surely in $(-1,1),$ in which case $Z^n\to 0.$ Thus $Y_n = Z^n \to 0$ almost surely.