Can the distance between 2 non-empty sets be infinite?

Question

Intuitively I would immediately assume no, but that's not how things usually work in math and considering there are different kinds of infinities I haven't been able to find the answer.

Here's my definition of the distance between 2 sets:

$d(A,B) = \inf{\{||\vec a - \vec b||:\vec a \in A, \vec b \in B\}}$

Answer 1

The distance between two sets of the same metric space is defined as:

$$d(A,B) = \inf_{x\in A,\ y\in B} d(x,y)$$

That means that if $x\in A$ and $y\in B$ then $d(x,y) \geq d(A,B)$.

Now, $d(x,y)$ is always finite in a metric space so $d(A,B)$ must be too.

Answer 2

Let $x$ be an element of $A$ and let $y$ be an element of $B$. We know that $\|x-y\|$ is a real number $r$ and so $d(A,B)$ must be at most $r$ by the definition of $d(A,B)$, hence $d(A,B)$ is finite.