What is the *exact* definition of a bounded subset in a metric space? I see quite a lot of different definitions of a bounded space. For instance, from nLab:

Let $E$ be a metric space. A subset $B⊆E$ is bounded if there is some real number $r$ such that $d(x,y)<r$ for all $x,y∈B$.

From Wiki:

A subset $S$ of a metric space $(M, d)$ is bounded if it is contained in a ball of finite radius, i.e. if there exists $x$ in $M$ and $r > 0$ such that for all $s$ in $S$, we have $d(x, s) < r$.

If I understand it correctly, the first definition requires that the origin of the "open disc" must be inside the subset, whereas the second definition does not have this restriction. Are these definitions somehow the same, or are they different? If not which one is correct?
The motivation for this question is because I want to understand what the Heine-Borel Theorem means.

Assuming the subset is nonempty, the two definitions are equivalent. Each implies the other.
