Rudin's 'Principle of Mathematical Analysis' Problem 7.12

Question

Suppose $g$ and $f_n$ ($n = 1,2,\ldots$) are defined on $(0,\infty)$, are Riemann-integrable on $[t,T]$ whenever $0 < t < T < \infty$, $|f_n| \leq g$, $f_n \rightarrow f$ uniformly on every compact subset of $(0,\infty)$, and

$$ \int_0^{\infty}\,g(x)\,dx < \infty $$

Prove that

$$ \lim_{n \rightarrow \infty}\,\int_0^{\infty}\,f_n(x)\,dx = \int_0^{\infty}\,f(x)\,dx $$

I appreciate all your comments, thanks.

Answer 1

Rather than writing my solution on here, I think its best if I gave you the official answer from the solution manual.

The solution manual of "Walter Rudin's Principles of Mathematical Analysis" can be found here http://minds.wisconsin.edu/handle/1793/67009

I have also posted the answer to your question as an image below.

The images above contain the solution. Let me know if you need further help.