I am reading Real Analysis by Stein & Shakarchi. On page 11, he gives a proof of how the exterior measure of a closed cube is equal to its volume. I follow the entire argument, but there is one part that's been bugging me. Here it is.
If we take $Q$ to the closed cube of interest, it is clear that $m_*(Q) \leq v(Q)$ where $m_*(Q)$ is the exterior(outer) measure. Then, it suffices to show that $v(Q) \leq m_*(Q)$.
The author says, consider any arbitrary covering $Q \subset \cup_{j=1}^{\infty}Q_j$ by cubes. Then, he argues that it suffices to prove that
$$ v(Q) \leq \sum_{j=1}^{\infty} v(Q_j)$$
That's the part I don't understand. Why does it suffice to prove this? $m_*(Q)$ is defined as the infimum over all countable coverings by cubes, so I don't think it makes sense to say that taking any arbitrary covering and showing that $v(Q)$ is bounded above by that expression proves that $v(Q)$ is also bounded above by the infimum. Where am I making a mistake?