Inverse image a of bounded set is bounded

Question

Knowing that the inverse image of a compact set is compact, prove that the inverse of a bounded set is bounded. ($f:\mathbb{R}^n\rightarrow \mathbb{R}^n$ continuous and $\mathbb{R}^n$ is equipped with the usual distance norm-1) My problem is that I don't know what to do and also , is it true that a set is bounded if it is contained inside an open ball ?

Answer 1

Yes, a set $A$ is bounded if there exists an open ball that contains it. But notice that open balls are also contained in closed balls, so any bounded set is also contained in a closed ball $B$. What do you know about $f^{-1}(B)$, and how do $f^{-1}(A)$ and $f^{-1}(B)$ relate to one another?

Answer 2

A subset of $\mathbb{R}^n$ is bounded iff it is a subset of a compact set. This follows from Heine-Borel, essentially. This immediately implies what you want: $A \subseteq K$ implies $f^{-1}[A] \subseteq f^{-1}[K]$.