What is the definition of Big $O$ notation in metric space

Question

The definition of $a_n=O(1)$ in $\mathbb{R}$ is that $a_n$ is bounded by a positive constant $M$ for every $n$, how to extend $O(1)$ to metric space? if $a_n$ is a sequence in a metric space $(D, d)$，then the definition of $a_n=O(1)$ is that there exists a compact set $K\subset D$ such that $a_n\in K$ for every $n$?

Answer 1

What you said is correct. But recall that compact sets in metric spaces are closed and bounded. So, you could simply say that a sequence $a_n$ is compact if $d(a_n, 0) \leq M$ for some constant $M \in \mathbb{R}_{>0}$. This way you are in complete correspondence with the case of $D = \mathbb{R}$.