Example to show the distance between two closed sets can be 0 even if the two sets are disjoint [duplicate]
Solution 1:
Let $A = \mathbb N$ and let $B = \left\{n+\frac{1}{2n} :n\in \mathbb N\right\}$. Then A and B are closed and disjoint, but $$\inf \{|a−b|:a \in A,b \in B\} = \inf \left | \frac{1}{2n}\right| = 0$$
Solution 2:
Consider the sets $\mathbb N$ and $\mathbb N\pi = \{n\pi : n\in\mathbb N\}$. Then $\mathbb N\cap \mathbb N\pi=\emptyset$ as $\pi$ is irrational, but we have points in $\mathbb N\pi$ which lie arbitrarily close to the integers.