Let $(X,d)$ and $(Y,\rho)$ be metric spaces, and let $f: X \to Y$ be a uniformly continuous function. If $A \subset X$ is bounded, must $f(A) \subset Y$ be bounded?
It is clear to me that in metric spaces that satisfy the Heine-Borel property, such as $\mathbb{R}^{n}$, the answer to this question is yes. However, I can see no reason why this should hold for arbitrary metric spaces. Any ideas? Thanks!