Show that if there exists 0 such that an for all n then

Question

Throughout, A, B, {A_n, n ≥ 1}, and {B_n, n ≥ 1} are subsets of ?.

Let An, n ≥ 1, be Borel sets on the Lebesgue space ([0, 1], F(0, 1), λ). Show that, if there exists η>0, such that λ(An) ≥ η for all n, then there exists at least one point that belongs to infinitely many sets An.