What is the definition of a bounded sequence in a metric space?

Question

Let $(X,d)$ be a metric space where $X$ is some set and $d : X \times X \rightarrow \Bbb R$ is a metric. And let $(x_n)$ be a sequence.

What is the definition of a bounded sequence in a metric space?

I would assume it's something of the form $\exists M , |x_n| \lt M$ for all $n$. But $|x|$ might not have meaning depending on what metric space is used.

Answer 1

A sequence is bounded if it is contained in a ball, so $\exists x^* \in X, M > 0$ such that $(x_n)\subseteq B_M(x^*)$. In some sense, you replace $|x_n| <M$ with $d(x^*,x_n) < M$.

Answer 2

If you are working in an abstract metric space, without any assumed vector space structure, there may not be a special point identified as zero. So you won't be able to define an absolute value via $|x| := d(x,0)$, since there is no zero point in $X$. But zero isn't all that special anyway when determining the boundedness of subsets of $X$. You can center balls about any point in $X$. So, you can say the sequence $(x_{n})$ is bounded if it is contained in a ball centered about some point $x^{\ast}$ in $X$; that is, there is an $x^{\ast} \in X$ and a $M > 0$ such that $d(x_{n},x^{\ast}) < M$ for all $n$.