I got stuck in Problem 41 of Pugh's book in Real Analysis (2nd edition) ,in which he asks to prove unit ball with any norm in Rn is compact. Since the unit ball is compact if it is closed and bounded by Heine-Borel Theorem.I don't know how to prove this ball is bounded with respect to Euclidean metric. Please help.

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    $\begingroup$ Can you prove that all norms on $\mathbb{R}^n$ are equivalent? $\endgroup$ – Daniel Fischer Oct 30 '16 at 14:26
  • $\begingroup$ Can you explain it clearer ? I don't know what "equivalent" does mean. $\endgroup$ – Viet Hoang Oct 31 '16 at 1:53
  • $\begingroup$ Wikipedia says "Two norms $\left\|\cdot \right\|_{\alpha }$ and $\left\|\cdot \right\|_{\beta }$ on a vector space $V$ are called equivalent if there exist positive real numbers $C$ and $D$ such that for all $x$ in $V$ $C\left\|x\right\|_{\alpha }\leq \left\|x\right\|_{\beta }\leq D\left\|x\right\|_{\alpha }$. ... Equivalent norms define the same notions of continuity and convergence and for many purposes do not need to be distinguished". See also about topological structure of normed vector spaces. $\endgroup$ – Alex Ravsky Nov 11 '16 at 19:50

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