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.