How to see equivalence of definitions of Lebesgue (outer) measure?

Question

By definition, the Lebesgue (outer) measure of a set $A \subset \mathbb{R}$ is given by$$m^*(A) = \inf_{\bigcup I_i \supset A} \sum_i m(I_i),$$where the infimum is taken over all countable collections of intervals $I_i$ such that $A \subset I_i$. How do I see that the following definition is equivalent:$$m^*(A) = \inf_{G \supset A} m(G),$$where the infimum is taken over all open sets $G \supset A$?