In the book measure theory and fine properties of functions of Evans and Gariepy, i'm trying to understand the proof of theorem 2.2 (ii) which states that $\mathcal{L}^1 = \mathcal{H}^1$. Here the Hausdorff measure is defined with the normalized constant, so that $$\mathcal{H}_\delta^1 = \inf \bigg\{ \sum_{j=1}^\infty diam C_j: A \subset \bigcup_{j=1}^\infty C_j, diam C_j \leq \delta\bigg\} $$
I understand the first part of the proof which proves that $\mathcal{L}^1 \leq \mathcal{H}^1(A)$ so I will just type the second one.
Let $\delta > 0$ and $C_j$ a covering $A$ such that $diam C_j \leq \delta$. Consider $I_j = [k\delta,(k+1)\delta]$, then $diam C_j \cap I_k \leq \delta$ and $$\sum_{k=1}^\infty diam(C_j \cap I_k) \leq diam (C_j) $$
Why is this inequality true? I tried to prove it but with no success. Then the proof follows with
$$\mathcal{L}^1 = \inf\bigg\{ \sum_{j=1}^\infty Diam C_j :A \subset \bigcup_{j=1}^\infty C_j\bigg\} \geq \inf\bigg\{ \sum_{j=1}^\infty \sum_{k=1}^\infty Diam C_j \cap I_k : A \subset \bigcup_{j=1}^\infty C_j \bigg\} \geq \mathcal{H}_\delta^1 $$
I don't understand the last inequality, where does it come from? Thanks in advanced!
Hints: For the first question use the fact that if $E$ is a bounded set of real numbers then the diameter of $E$ equals the Lebesgue measure of the interval from $\inf E$ to $\sup E$.
For the second question use the fact that $A \subset \bigcup_{j=1}^\infty C_j$ implies $A \subset \bigcup_{j,k=1}^\infty (C_j\cap I_k)$ (and use the definition of $\mathcal H^{1}_\delta$