Let $f: [a,b] \rightarrow \mathbb{R}$ be a Lipschitz function.
Let $\epsilon > 0$.
Let $E$ be a set of measure zero. There exists countable, bounded, open intervals with the form $I_n=(x_n, y_n)$ such that $E \subseteq \bigcup\limits_n I_n$ and for some $x_k, y_k \in I_k$, then $|f(x_k)-f(y_k)| \leq c|x_k -y_k|$. I believe this sufficiently incorporates the definition of Lipshitz.
Now I want to cover $E$ with the intervals such that $\sum\limits_n l(I_k) < \frac{\epsilon}{c}$.
So, for every n we pick $I_{n}^{'} \subset f(I_n)$ which are open, bounded, and countable such that $l(I_{n}^{'}) \leq c l(I_{n})$. The collection $\{I_{n}^{'}\}$ is a cover for $f(E)$ and $\sum\limits_n l(I_{n}^{'}) \leq \sum\limits cl(I_{n})< c\frac{\epsilon}{c} =\epsilon$.