The requirement that $\displaystyle \left( {\forall m,n} \right)\left[ {[a_n ,b_n ] \cap [a_m ,b_m ] \ne \emptyset } \right]$ implies that $\displaystyle \left( {\forall m,n} \right)\left[ {a_n \leqslant b_m } \right]$.
So $\displaystyle A = \left\{ {a_n :n \in \mathbb{Z}^ + } \right\}$ has a least upper bound. Call it $\displaystyle \alpha$.
Show that $\displaystyle \alpha \in \bigcap\limits_n {\left[ {a_n ,b_n } \right]} $.