Riemann integral probelm

Question

Let $f: [a,b] \rightarrow \mathbb{R}$ be continuous. Suppose $(P_n)$ is a sequence of partitions with mesh$(P_n) \rightarrow 0$ as $n \rightarrow \infty$. Prove that $\int_{a}^{b} f(x) dx = \lim_{n \to \infty} U(f, P_n) = \lim_{n \to \infty} L(f, P_n)$.

Definitions:

• mesh$(P) = \max\{|x_i - x_{i-1}| : i = 1,2,3,\ldots, n\}$ where $P$ is any partition of $[a,b]$.

• $U(f , P) = \sum_{i=1}^{n} M_i(x_i - x_{i-1})$ where $M_i = \sup\{f(x) : x \in [x_{i-1}, x_i]\}$

• $L(f , P) = \sum_{i=1}^{n} m_i(x_i - x_{i-1})$ where $m_i = \inf\{f(x) : x \in [x_{i-1}, x_i]\}$

• $U(f) = \inf\{U(f , P) :$ P is a partition of $[a,b]\}$

• $L(f) = \sup\{L(f , P) :$ P is a partition of $[a,b]\}$

• $f$ is integrable on $[a,b]$ iff $U(f) = L(f)$.

I am pretty lost on how to prove this. I think we can prove this statement if we can first show that $\lim_{n \to \infty} [U(f, P_n) - L(f, P_n)] = 0$. From there, if we can show $\lim_{n \to \infty} U(f, P_n) = U(f)$ and $\lim_{n \to \infty} L(f, P_n) = L(f)$, then since $f$ is continuous on $[a,b]$ it must also be integrable on $[a,b]$ and hence, $U(f) = L(f)$. I am also confused on where to use the mesh of $P_n$. Again, since $f$ is continuous on $[a,b]$, we know that $f$ is uniformly continuous on $[a,b]$. So do we somehow use the mesh of $P_n$ to find an $N \in \mathbb{N}$ such that $x_i - x_{i-1} < \delta$? Any help is appreciated!

Answer 1

This is true for any Riemann integrable function:

Let $K=\max f$ on $[a,b]$, and choose a partiton $P$ such that $U(f,P) - L(f,P) <\epsilon.\ $ Suppose $P$ has $N$ points. Now let $P_n$ be one of the partitions in the sequence with mesh $\delta<\epsilon /NK.$ Consider $Q=P\cup P_n$ which is a refinement of $P_n$. Certainly

$\tag1U(f,Q) - L(f,Q) <\epsilon. $

Furthermore, since $Q$ is obtained by adding $N$ points to $P_n,$ we also have

$\tag2 U(f,P_n)-U(f,Q)<2NK\delta <2\epsilon.$

The same reasoning shows that

$\tag3 L(f,Q)-L(f,P_n)<2\epsilon.$

Thus, using $1), 2)$ and $3), $

$U(f,P_n)-L(f,P_n)=U(f,P_n)-U(f,Q) + U(f,Q) - L(f,Q) + L(f,Q) - L(f,P_n)<5\epsilon, $

and this completes the proof.