I'm tagging this question homework because I'm more interested in hints than in complete solutions. First let us give a definition.
Definition Let $X$ be a metric space. For all $F \subset X, \rho \ge 0$, let $F_\rho$ be the union of all closed $\rho$-balls centered at some point of $F$. For all compact $C, D$, call Hausdorff distance the quantity
$d_H(C, D)=\inf(\rho \ge 0 \mid C \subset D_\rho, D \subset C_\rho).$
(Wikipedia on Hausdorff distance).
Now take two simple smooth loops $\gamma_1, \gamma_2\colon[0, 2\pi]\to \mathbb{R}^2$ and denote with $D_1, D_2$ the compact subsets of $\mathbb{R}^2$ they encircle. The problem states that
$d_H(D_1, D_2) \le \lVert \gamma_1 - \gamma_2 \rVert_{\infty}.$
How to prove this?
I've thought about it quite a little bit. We are here asked to prove that, for every $\xi \in D_1$, there is some $\eta \in D_2$ such that $\lvert \xi - \eta \rvert \le \lVert \gamma_1-\gamma_2\rVert_{\infty}$ (the claim will follow by interchanging $D_1$ and $D_2$). In other words, it is to be shown that we can always travel from a fixed point in $D_1$ to somewhere in $D_2$ in less than $\lVert \gamma_1-\gamma_2 \rVert_{\infty}$ units of distance. But I must admit that I can't show this. Can someone lend me a hand?