I would like to prove the following statement: Let $S,T \subseteq \mathbb{R}^{n}$ be closed sets with $S \cap T = \emptyset$, at least one of which is bounded. Then there exist $x \in S$ and $y \in T$ such that $d(x,y) \leq d(\hat{x},\hat{y}) \text{ for all } \hat{x} \in S, \hat{y} \in T,$ where $d(\cdot,\cdot)$ is the Euclidean distance. Should be simple, but couldn't find the proof immediately. Could anyone help me please? Thanks a lot!
Tanja