# real analysis – Proving exterior measure of a cube is equal to its volume

This is from Stein and Shakarchi’s Measure Theory. I don’t understand why we must use this whole $$epsilon$$-proof (below the black line). Wouldn’t the following much simpler proof show $$|Q| leq sum_{j=1}^infty |Q_j|$$?

We’ve previously proven a lemma that states if $$R, R_1, …, R_N$$ are rectangles, and $$R subseteq cup_{j=1}^infty R_j$$, then
$$|R| leq sum_{j=1}^N |R_j|.$$

This is for finite $$N$$, but since each $$|R_j|$$ is positive, this must hold for an infinite collection of rectangles as well. Therefore, the claim that
$$|Q| leq sum_{j=1}^infty |Q_j|$$

follows immediately.