Definition of bounded in a metric space - confirmation Is the following definition of a bounded metric space correct?

$(M,d)$ is bounded if $\exists a \in M, r > 0$ such that $M = B(a,r)$.

Looking around on the internet I instead see $M \subset B(a,r)$.
 A: The two definitions are equivalent. In particular, since a ball is defined as
$$B(a,r)=\{x\in M:d(x,a)<r\}$$
it is trivial that we have $B(a,r)\subseteq M$ for any $a$ and $r$. Knowing this, the statement that $M\subseteq B(a,r)$ implies that $B(a,r)=M$ since $\subseteq$ is an antisymmetric relation.
However, one might note that if you want to define a bounded subset $S\subseteq M$, then you would write $S\subseteq B(a,r)$ rather than $S=B(a,r)$, since the ball would be taking place in $M$ rather than intrinsically $S$.

The definition $M\subseteq B(a,r)$ is a good definition for a metric space or subset thereof being bounded. This coincides with the intuition people want to capture by boundedness, though it is equivalent to other definitions. Moreover, in the definition $M=B(a,r)$, one could easily forget that the ball on the right hand side of the equation must be taken with respect to $M$ and not to some larger space, where writing $M\subseteq B(a,r)$ does not allow one to make such a mistake.
