Let $A_1$, $A_2$, ..., be independent events. Show that $P\left(\bigcup\limits_{j=0}^\infty A_{i_j}\right)=1$ for every subsequence $1\le i_1\le i_2\le$ ... of integers if and only if $\liminf_{n\to \infty} P(A_n) \gt 0$.
I don't know how to deal with these subsequences (in fact, we have "for every subsequence", which makes it more difficult for me). Also, I have mostly encountered problems dealing with limsup. So this problem appears to be new altogether. Can someone please guide me through this?
Assume that $\liminf_{n\to\infty}P(A_n)=0$. Then for each $n$ we can pick $i_n$ (as an increasing sequence) such that $P(A_n)<3^{-n}$. Then $P(\bigcup_{j=1}^\infty A_{i_j})\le \sum_{j=1}^\infty 3^{-j}=\frac12<1$.
On the other hand assume $\liminf_{n\to\infty}P(A_n)=c>0$. Then for any finite subsequence of length $n$ we have $$P(\bigcup_{j=1}^nA_{i_j})=1-P(\bigcap_{j=1}^n\overline{A_{i_j}})=1-\prod_{j=1}^ n(1-P(A_{i_j}))\ge1-(1-c)^n$$ Taking the limit as $n\to \infty$, we see that $P(\bigcup_{j=1}^\infty A_{i_j})=1$.