So I have a sequence of riemann (darboux specifically) integrable real valued functions $(f_n)$ defined on $[a,b] \subset \mathbb{R}$ that converges uniformly to some $f$. I would like a hint (or two even) on how to show this $f$ is integrable. I split this problem up into two parts, namely first proving that $f$ is bounded, and then later proving that $f$ is integrable. So for the first part I have that:
Recall that each $f_n$ is bounded on $[a,b]$, so we have that for each $n$, $f_n([a,b])$ has some supremum in $\mathbb{R}$ (call it $M$) and that:
$$
f_n(x) \leq M \text{ for any $x\in [a,b]$}
$$
A similar statement can be made about the lower bound. Fix some $\epsilon>0$. We know by the uniform convergence of $f_n$ to $f(x)$ that, there exists $N \in \mathbb{N}$ with the property that for any $n \geq N$ we have:
$$
|f_n(x) – f(x)|<\epsilon \text{ for all $x\in [a,b]$}
$$
So we see that:
$$
f(x)< f_n(x) + \epsilon \leq M +\epsilon \text{ for all $x\in [a,b]$}
$$
So $f$ is bounded.
I am not entirely sure how to proceed from here. I am fairly certain I need to satisfy the Riemann integrability condition, i.e show that for any $\epsilon$ there exists a partition $P$ of $[a,b]$ such that:
$$
U(f,P) – L(f,P) < \epsilon
$$
I have a feeling I need to pick some special epsilons and do some tricky manipulations, but I was hoping for some guidance in the right direction.
Best Answer
HINT: Write \begin{equation*} \begin{aligned} U(f,P) - L(f,P) &= U(f,P) - U(f_n,P) + U(f_n,P) - L(f_n,P) + L(f_n,P) - L(f,P) \\ &\leq \vert U(f,P) - U(f_n,P) \vert + U(f_n,P) - L(f_n,P) + \vert L(f_n,P) - L(f,P) \vert. \end{aligned} \end{equation*}
As an aside, it's worth noting that once you prove this, you can also use uniform continuity to prove that $$\lim_{n\to\infty}\int_a^b f_n(x)dx = \int_a^b f(x)dx,$$ i.e. you can interchange the order of the limit and the integral.