Exercise 7.18

Question

Exercise 18: 
    Let $\{f_n\}$ be a uniformly bounded sequence of functions which are Riemann-integrable on $[a,b]$, and put $$F_n(x)=\int_a^x f_n(t)\,dt\quad\quad(a\le x\le b).$$ Prove that there
    exists a subsequence $\{F_{n_k}\}$ which converges uniformly on $[a,b]$.

analambanomenos · Accepted Answer

\def\eps{\varepsilon}
\def\ints{\mathbb{Z}}

This follows from Theorem 7.25 if we show that $\{F_n\}$ is pointwise-bounded and equicontinuous on $[a,b]$. Let $|f_n|\le K$ on $[a,b]$. Then for all $n$ $$\big|F_n(x)\big|\le\int_a^x\big|
    f_n(t)\big|\,dt\le K(x-a),$$ so $\{F_n\}$ is pointwise-bounded on $[a,b]$. And if $\varepsilon>0$ and $x<y$ are points of $[a,b]$ such that $y-x\le\varepsilon/K$, then $$\big|F_n(x)-F_n(y)\big|
    \le\int_x^y\big|f_n(t)\big|\,dt\le K(y-x)\le\varepsilon,$$ showing that $\{F_n\}$ is equicontinuous on $[a,b]$.

Exercise 7.18

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Exercise 7.18

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