I was trying to solve the following problem:
This is what I have tried. Please tell me if my attempt is correct:
Is my above arguments correct?
2026-03-29 08:35:33.1774773333
$C^1[0,1]$ is Banach Space
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I think your last argument can be simplified. You have $f_n \to f, \,f_n'\to g$ uniformly on $[0,1].$ For any $x\in [0,1]$ we have
$$f_n(x) - f_n(0) = \int_0^x f_n'$$
by the FTC. But the left side $\to f(x) - f(0)$ and the right side $\to \int_0^x g.$ Thus $f\in C^1,$ by the FTC.