Consider the unit ball [tex]B_1=\{f:\rho_u(f,0)\leq 1\}[/tex] in the metric space [tex](C[0,1],\rho_u)[/tex] where [tex]\rho_u(f,g)=sup\{\forall x(|f(x)-g(x)|)\}.[/tex] Show that there exists a sequence [tex]g_n\in B_1[/tex] such that NO subsequence of [tex]g_n[/tex] converges in [tex]\rho_u[/tex].(adsbygoogle = window.adsbygoogle || []).push({});

I want to know if what I'm doing is right. Suppose I define a sequence of functions that converge to a function OUTSIDE of C[0,1]=(the space of all continuous functions on 0,1), then any subsequence of such a function would not converge, right?

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# Homework Help: Unit Ball

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