Rudin page 152 Theorem 7.17: Suppose $\{f_n\}$ a sequence of functions, differentiable on $[a,b]$ and such that $\{f_n(x_0)\}$ converges for some point $x_0$ on [a,b]. If $\{f_n'\}$ converges uniformly on [a,b], then $\{f_n\}$ converges uniformly on [a,b] to $f$ and $f'(x)=lim_{n \to \infty} f_n'(x)$ with $a\le x \le b$.
Remark: if the continuity of the function $f′_n$ is assumed in addition to the above hypothesis, then a much shorter proof can be based on Theorem 7.16 and the fundamental theorem of calculus.
Theorem 7.16 states: Let $\alpha$ be monotonically increasing on [a,b]. Suppose $f_n$ is in Reimann(alpha) on [a,b], for n=1,2,3,…. and suppose $f_n \to f$ uniformly on [a,b]. Then f is in Reimann(alpha) on [a,b] and $$\int_a^bfd\alpha = \lim_{n \to \infty} \int_a^bf_nd\alpha .$$
Prove the Remark.
I think I have a good enough understanding of theorem 7.17 but I am very interested in the proof of the remark to help gain a better understanding.I cannot seem to make the connection. Would anyone be able to point me in the right direction?
Best Answer
The functions $f_n$ being continuous, the fundamental theorem of calculus tells us that $$ f_n(x) = f_n(x_0) + \int_{x_0}^xf_n'(t)\,dt $$ for $x \in [a,b]$ (this is theorem 6.21 in Rudin). Suppose that the functions $\{f_n'\}$ converge uniformly on $[a,b]$ to a (continuous) function $h$. Then define $$ f(x) := f(x_0) + \int_{x_0}^x h(t)\,dt. $$ By theorem 7.16 in Rudin, we know that $$ \int_{x_0}^x h(t)\,dt = \lim_{n \to \infty} \int_{x_0}^x f_n'(t)\,dt = \lim_{n \to \infty} (f_n(x) - f_n(x_0)) $$ where the last equality is again the fundamental theorem of calculus. This shows that indeed $$ f(x) = \lim_{n \to \infty} f_n(x). $$ But since $h$ is continuous (this is because we supposed the $f_n$'s to be of class $C^1$), we can again invoke the fundamental theorem of calculus to say that $$ f(x_0) + \int_{x_0}^x h(t)\,dt $$ is a differentiable function on $[a,b]$ with derivative h(x) (this is theorem 6.20 in Rudin). In other words, $f(x)$ is differentiable with $f'(x) = h(x) = \lim_{n \to \infty} f_n'(x)$, as we wanted.