Every sequence of sets have coverging subsequence

Question

Is that true, that every sequence of sets have coverging subsequence? We say that sequence of sets $A_1, A_2, A_3, ...$ coverging iff ${\limsup}_{n \to \infty} A_n = {\liminf}_{n \to \infty} A_n$

Answer 1

Hint: consider $$A_n = \left\{ f \in \{ 0, 1 \}^{\mathbb{N}} : f(n) = 1 \right\}.$$