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Suppose ${A_n}$ is a sequence of events.

If $P(A_n)\to 1$ as $n\to\infty$,prove there exists a subsequence ${n_k}$ tending to infinity such that $$P(\cap_kA_{n_k})>0$$

The textbook gives a hint :(Using Borel-Cantelli lemma). here is my proof:

since $P(A^c_n)\to0$,there is a subsequence ${n_k}$ s.t. $P(A_{n_k}^c)<\frac{1}{3^k}$ ,so by subadditivity there is $P(\cup_k A^c_{n_k})<\frac{1}{2}$ .So $P(\cap _kA_{n_k})>0$.

The problem is why there is nothing to do with Borel-Cantelli lemma?

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I think your argument is correct. The book may use the Borel-Cantelli rather than subadditivity. That is ,there's a subsequence that $\sum_{n_k} P(A_{n_k}^c)<\infty$. Thus $P(\limsup A_{n_k}^c)=0$. Which means, $$P\left(\bigcup_{j=1}^\infty\bigcap_{k=j}^\infty A_{n_k}\right)=1.$$ Let $E_j=\bigcap_{k=j}^\infty A_{n_k}$ and we know that $E_j\subset E_{j+1}\subset\cdots$. Then there's a $j$ such that $P(E_j)>0$, which is the desired result.

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