Covering of Intervals
Here is a disproof. Take any $\epsilon,\delta > 0$. We construct $([a_j,b_j])_{1 \le j \le n}, ([c_j,d_j])_{1 \le j \le n}$ such that:
(1) $|a_j-c_j| \le \delta$ and $|b_j-d_j| \le \delta$ for each $j$.
(2) $\sum_{j=1}^n |(a_j-c_j)-(b_j-d_j)| = 0$.
(3) $|\cup_{j=1}^n [a_j,b_j]| = \epsilon$.
(4) $|\cup_{j=1}^n [c_j,d_j]| = \sqrt{\epsilon \delta n}$.
Take any $L \in \mathbb{N}$, and let $R = \frac{L\delta}{\epsilon}$ and $n = RL$. (It's not a big deal to assume $R \in \mathbb{N}$). For $0 \le l \le L-1$ and $1 \le r \le R$, let $[a_{lR+r},b_{lR+r}] = [l,l+\frac{\epsilon}{L}]$ and $[c_{lR+r},d_{lR+r}] = [l-\delta+(r-1)\frac{\epsilon}{L},l-\delta+r\frac{\epsilon}{L}]$.