A problem related to limsup and liminf.

Question

Suppose $a_n >0$ and $b_n <1$ and lim $a_n=0$, lim $b_n =1$ and $A_n =\{x: a_n \leq x <b_n\}$ then find limsup$ (A_n)$ and liminf$(A_n) $.

As $a_n$ goes to 0 and $b_n$ goes to 1, How can I use this condition to find limsupA_n and liminf A_n, I got stuck.

Answer 1

$(A_n)_n$ is an increasing sequence of sets, thus it converges to $\cup_n A_n=\liminf A_n=\limsup A_n$ and $\cup_nA_n = [0,1).$