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Compact subset of R^n 
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#1
Mar508, 02:56 AM

P: 2

I am struggling to prove the following: Let E be a compact nonempty subset of R^k and let delta = {d(x,y): x,y in E}. Show E contains points x_0,y_0 such that d(x_0,y_0)=delta.



#2
Mar508, 03:17 AM

P: 2,023

[tex] \Delta := \sup\{d(x,y)\;\;x,y\in E\}\; ? [/tex] It is convenient to consider a function [itex]d:E\times E\to\mathbb{R}[/itex], and use some basic topological results, or their immediate consequences. For example: The Cartesian product of compact sets is a compact set. In metric spaces compact sets are sequentially compact. The distance function d is continuous. Continuous mappings map compact sets into compact sets. The HeineBorel Theorem. Just put pieces together! 


#3
Mar508, 11:57 AM

P: 2

\Delta but not sure how to show it. We have not talked about continuous functions. We have only studied sequences so far. thanks 


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