1.I.1F

Let $E$ be a subset of $\mathbb{R}^{n}$. Prove that the following conditions on $E$ are equivalent:

(i) $E$ is closed and bounded.

(ii) $E$ has the Bolzano-Weierstrass property (i.e., every sequence in $E$ has a subsequence convergent to a point of $E$ ).

(iii) Every continuous real-valued function on $E$ is bounded.

[The Bolzano-Weierstrass property for bounded closed intervals in $\mathbb{R}^{1}$ may be assumed.]

*Typos? Please submit corrections to this page on GitHub.*