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I have this here metric space problem which caused me some trouble:

[tex]S \subseteq \mathbb{R}^n[/tex] then the set

[tex]\{ \| x - y \| \ | y \in S \} [/tex] has the infimum

[tex]f(x) = \{ \| x - y \| \ | y \in S \}[/tex]

where f is defined [tex]f: \mathbb{R}^n \rightarrow \mathbb{R}[/tex]

I have two problems here which I'm unable to solve:

(a) show, if S is a closed set and [tex]x \notin S [/tex] then [tex]f(x) > 0[/tex] ????

(b) show, if S is a closed set, then [tex]S = \{ x \in \mathbb{R}^n | f(x) = 0\}[/tex] ???

I need to hand this in tomorrow, and I have been strugling this these two problems the last week, therefore I would very much appriciate if anybody could give me an idear on how to solve the two problems above.

God bless,

Best Regards,

Fred

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# Homework Help: HELP metric space problem

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