 Quote by Dick
You have that backwards. ||fn-fm||_E<epsilon implies ||f'n-f'm||_infinity<epsilon.
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Really? I don't see why this is so.
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Can you use the fact the difference in derivatives of fn and fm is small to prove the difference between fn and fm is small? Hence that fn(x) is a cauchy sequence for each x?
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But what good will that do?
So here is the interpretation of the assignment in my eyes:
Given a Cauchy sequence [tex](f_n)_n \in\ E[/tex] prove that [tex]||f_n-f||_E \rightarrow 0 [/tex] and that f is in E.
So we have:
[tex] ||f'_n -f'_m||_{\infty} < \epsilon\ \forall m,n \geq N [/tex]
and we want: [tex] ||f'_n -f'||_{\infty} \rightarrow 0\ \forall n \geq N [/tex]
Is this correct?