Showing Non-Convergence of a Sequence in B_1

[Treadstone 71]

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

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?

 

[AKG]

Right...

 

[Treadstone 71]

But at no point did I use the fact that the function I defined doesn't converge in $$\rho_u$$.

 

[AKG]

Well you're missing a lot of facts. I figured you would fill in the details, and you just wanted to know if your idea would work. So yes, it will work. C[0,1] is a subspace of B[0,1], the space of all bounded real-valued functions on [0,1]. You can give B[0,1] the same metric. Then there are some more easy details to work out, but you can do it. In fact, I don't know if you have to regard C[0,1] as a subspace of any other space, you can try to prove more directly that no subsequence converges.

 

[Treadstone 71]

Great. Thanks.