Outer measure countable subadditivity proof confusion

Question

Using the definitions:

Why is it assumed that there exist intervals such that the highlighted part is true?

I thought $|A_k|$ is supposed to be in the infimum of $\sum_{j=1}^{\infty} l(I_{j,k})$ over some sequence.

Then how is it possible that this inequality could be true?

Thank you.

Answer 1

Recall that on $\Bbb R$ $$\alpha=\inf A\iff \forall\varepsilon>0\ \exists a\in A\text{ such that } a\le \alpha+\varepsilon.$$

Just apply this with $\alpha=|A_k|$, $\varepsilon=\dfrac{\varepsilon}{2^k}$ and $A=\big\{\sum_{j=1}^\infty \ell(I_{j,k}):\ A_k\subset\bigcup I_k\big\}$