Show that if $m^∗(E) < ∞$ and there exist intervals $I_1, . . . , I_n$ such that $m∗E(∆(∪_{i=1}^{n}I_i))< ∞$, then each of the interval are finite

There exists an open set $U$ such that $E subseteq U$ and $m(U) <m^{*}(E)+1$. We can write $U$ as countable disjoint union of open intervals $(a_i,b_i)$. Now $bigcup_{i=1}^{n} (a_i,b_i)) subseteq (EDelta bigcup_{i=1}^{n} (a_i,b_i)) cup E$ so $m(bigcup_{i=1}^{n} (a_i,b_i))<infty$. This implies that each of the $I_i$'s is a finite interval.