About proof of "Exterior measure of a closed cube is equal to its volume".

Question

I suppose my question is related to my basic misunderstanding of something.

Part of the proof:

In the proof we see "Since $Q$ covers itself, $\textbf{we must have } m^*(Q)\leq |Q|$...". Then we prove $m^*(Q)\geq |Q|$ and finish the proof.

My question is about first argument. How can infinum of coverings be less than volume? Why we write $m^*(Q)\leq |Q|$? Why $\textbf{we must have}$ it?

Where is an example when we can say that $m^*(Q)< |Q|$?

Edit: And if there is not such example why we can not simply state at the very start of the proof that $m^*(Q)= |Q|$?

Edit: If it was something like that "Let $Q^.$ will be any covering of $Q$ then we must have $m^*(Q)\leq |Q^.|$." And it is ok with me. But in the proof we take precisely the $Q$.