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I have troubles with proving that the closure of a bounded set is bounded. Can somebody help me out with this problem? Thanks!

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  • $\begingroup$ What does it mean for the set to be bounded? $\endgroup$ – The very fluffy Panda May 17 '14 at 10:31
  • $\begingroup$ A is bounded if there exists a M>=0 (real) such that d(x,y)<=M for all x,y in A. $\endgroup$ – Roos Jansen May 17 '14 at 10:38
  • $\begingroup$ Well, now with this in mind. Can you a find a set that encloses A? $\endgroup$ – The very fluffy Panda May 17 '14 at 10:44
  • $\begingroup$ Yes, there exists a x0 and a r>0 such that B(x0,r) ensloses A. $\endgroup$ – Roos Jansen May 17 '14 at 11:31
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Let $M$ be a bounded set. We have to prove that $\sup_{x,y\in \overline{M}}d(x,y)$ is finite.

Take $x,y\in \overline{M}$; by definition of the closure, there exists $x'$, $y'\in M$ such that $d(x,x')\lt 1$ and $d(y,y')\lt 1$. We thus have, by the triangle inequality, $$d(x,y)\leqslant d(x,x')+d(x',y')+d(y,y')\leqslant 2+\sup_{u,v\in M}d(u,v).$$ We are done, since the RHS does not depend of $x$ or $y$.

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