TITLE: Example to show the distance between two closed sets can be 0 even if the two sets are disjoint
QUESTION [21 upvotes]: Let $A$ and $B$ be two sets of real numbers. Define the distance from $A$ to $B$ by $$\rho (A,B) = \inf \{ |a-b| : a \in A, b \in B\} \;.$$ Give an example to show that the distance between two closed sets can be $0$ even if the two sets are disjoint.

REPLY [35 votes]: 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 $$\inf \{|a−b|:a \in A,b \in B\} = \inf \left | \frac{1}{2n}\right| = 0$$

REPLY [12 votes]: 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.

REPLY [7 votes]: Let $A$ be the set of positive integers, and let $B$ be the set of all numbers of the form $n+1/n$, where $n$ ranges over the integers $\gt 1$.