On a property of a sequence of events

Let $\{A_n\}$ be a sequence of events and $\varlimsup\mathbf P(A_n)=p>0$. Then it is possible to choose for any $c<1$ a subsequence with the property: for any $k$, the probability of the intersection of arbitrary $k$ events of the subsequence is more than $cp^k$.
This statement becomes false if we replace the constant $c$ by a sequence $c_k\to1$.