Given a metric space $(X,d)$, I have the Hausdorff Metric defined as follows for two non empty subsets $Y,Z \subset X$:
$$d_H(Y,Z) = \max \{\sup_{y\in Y} d(y,Z), \sup_{z\in Z} d(z,Y)\}$$ $$\text{where }d(y,Z)=\inf_{z \in Z}d(y,z)\text{, and }d(z,Y)=\inf_{y \in Y}d(z,y)$$ I need to come up with an example where $d_H(Y,Z) = \infty$, with the fact that $Y,Z \subset X$ are closed.
My example:
Let $Y = \mathbb{R}, Z = \{0\}$. Then, $$d(0,\mathbb{R})=\inf_{y \in \mathbb{R}}d(0,y)=0$$ $$d(y,\{0\})=\inf_{z \in \{0\}}d(y,z)=d(y,0)=|y|$$ $$d_H(\mathbb{R},\{0\})=\max \{\sup_{y\in \mathbb{R}} d(y,\{0\}), \sup_{z\in \{0\}} d(0,\mathbb{R})\}=\max \{\sup_{y\in \mathbb{R}} |y|, 0\}=\infty$$
Am I correct?