= Solution
We prove the equivalence by a cycle of implications. First assume closedness and boundedness. Given a <sequence> $(x_j)$ in $E$, boundedness puts each coordinate in a fixed closed bounded real interval. Apply the one-dimensional <Bolzano-Weierstrass theorem> to its first coordinate to extract a convergent subsequence, then apply it to the second coordinate of that subsequence, and continue through the finitely many coordinates. The final subsequence converges coordinatewise to some $x\in\mathbb R^n$. It also converges in the <Euclidean norm>, since the sum of the squares of its finitely many coordinate differences tends to zero. Since $E$ is a <closed set>, this limit belongs to $E$. Thus \b[closedness and boundedness imply the required subsequence property]. No higher-dimensional compactness theorem has been assumed.
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