# probability convergence

$P(X_n) \to 1/2, P(Y_n) \to 1, P(Z_n) \to 1$ as $n \to \infty$. prove $P(X_nY_nZ_n) \to 1/2$ as $n\to \infty$. I'm stuck on this question, not sure how I should proceed. Any help would be greatly appreciated, thanks!

• Are there any independence condition on those events? – Xuqiang QIN Dec 16 '16 at 1:30
• There are no independence condition on them – Xixi Dec 16 '16 at 2:31
• What do you mean that $P(X_n) \to 1/2, n \to \infty$? Is this an a.s. convergence to the degenerate random variable $X = 1/2$? – Therkel Dec 16 '16 at 9:35
• @Therkel, the way I see it, $(X_n)_{n \in \mathbb N}$ is a sequence of sets not a sequence of random variables, and $\mathbb{P}(X_nY_nZ_n)$ means $\mathbb{P}(X_n,Y_n,Z_n)$ – Cettt Dec 16 '16 at 15:13
• What exactly is meant by $$\mathbb P\left(X_n,Y_n,Z_n\right)\quad \large \mathrm ?$$ – Math1000 Dec 17 '16 at 8:42