Analysis II | Part IB, 2003

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

(i) EE is closed and bounded.

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

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

[The Bolzano-Weierstrass property for bounded closed intervals in R1\mathbb{R}^{1} may be assumed.]

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