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Consider the set $F$ of functions from $[0 , 1]$ to $[0 , 1]$ with the metric $(f, g) → sup${$|f(x) − g(x)| x ∈ [0 , 1]$}. Let $C$ denote the collection of constant functions in $F$. Show that $∂C = C$.

To show that $∂C = C$ we have to show $∂C \subseteq C$ and $C \subseteq ∂C$.

I am able to show that $C \subseteq ∂C$ but I am unable to show that $∂C \subseteq C$!!

Please Help... Thank You!!

User8976
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  • For that it is enough to prove that $C$ is closed. – drhab Jan 16 '15 at 12:48
  • well that can be a way .....but can you help me to complete in the way i stared the proof!! – User8976 Jan 16 '15 at 13:03
  • If you can show that $f_n\rightarrow f$ for a sequence $(f_n)$ in $C$ implies that $f\in C$ then you are ready in proving that $C$ is closed. Then $\partial C\subseteq\overline{C}=C$. – drhab Jan 16 '15 at 13:14

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