Exercise 7.12

Fix $t>0$ and let $k(T)={\int }_t^T|f(x)|\mathit{dx}$. Then $k(T)$ is a monotonically increasing function which is bounded by ${\int }_t^{\infty }g(x)\mathit{dx}$. Hence by Theorem 3.14 (that theorem refers to sequences, but can be easily extended to this case), $k(T)$ converges as $T\to \infty$, and since $|{\int }_t^{T}f(x)\mathit{dx}|\le k(T)$, ${\int }_t^{\infty }f(x)\mathit{dx}$ converges. Similarly, ${\int }_t^{\infty }{f}_n(x)\mathit{dx}$ also converges.

For $t>0$ define the functions

The problem is to show that

By Theorem 7.11, this follows if we can show that $h_n(t)$ converges uniformly to $h(t)$ on $(0,\infty )$.

Let $𝜀>0$. Since ${\int }_0^{\infty }g(x)\mathit{dx}$ converges, there are numbers $0<T_1<T_2<\infty$ such that

Since $f_n$ converges to $f$ uniformly on $[T_1,T_2]$, there is a positive integer $N$ such that for $n>N$ and $T_1\le x\le T_2$ we have

Hence, for $0<t<T_1$ and $n>N$, we have

Similar arguments yielding similar results can be made for $t\ge T_1$. This shows that $h_n(t)$ converges uniformly to $h(t)$ on $(0,\infty )$.