I'm trying to prove that for every connected set $E\subset\mathbb{R}$, $H^1(E)$, the Hausdorff measure is bounded below by $\text{diam}(E)$.
In the answer there, it was suggested to use a Lipschitz mapping defined by $x\mapsto d(x,a)$. How does it prove the inequality $H^1(E)\ge\text{diam}(E)$?