Why are these two definitions of Lebesgue outer measure equivalent?

Question

I got two definitions of Lebesgue outer measure one is $$\mu^*(A)=\inf\Big\{\sum^{\infty}_{k=1}l(I_k):A\subset \bigcup_k I_k\Big\}$$ and the other is $$\mu^*(A)=\inf\Big\{\sum^{\infty}_{j=1} \operatorname{diam} C_j: A\subset \bigcup_j C_j\Big\}$$ In the second definition $C_j$ is any arbitrary subset of $\Bbb{R}$.

Why are these two equivalent?

Answer 1

If $I$ is an interval then $\mathrm{diam}(I) = \ell(I)$. Conversely, any set $C$ is contained in a closed interval $I$ satisfying $\ell(I) = \mathrm{diam} (C)$.