In Wikipedia:
A subset $S$ of $\mathbb{R}^n$ is bounded with respect to the Euclidean distance if and only if it bounded as subset of $\mathbb{R}^n$ with the product order.
More generally, I was wondering for a set which is both a metric space and partially-ordered set, when the boundedness of a subset wrt the metric and wrt the order agree, or just one implies the other not the other way around?
Thanks and regards!