Before I ask the question, I will post some defintions(They were written by GEdgar in an earlier post).
A partition of $\pi$ of $[a,b]$ is a sequence $\{a=a_0<a_1<\ldots<a_K=b\}$. The norm of the partition $||\pi||$ is defined as $\max_{1\le k\le K}|a_k-a_{k-1}|$.
Let $f$ be a bounded real function on $[a,b]$, and $\pi$ be a partition of $[a,b]$. We define $U(f,\pi)=\sum\limits_{k=1}^KM_k(f)(a_k-a_{k-1})$, where $M_k(f)=\sup\{f(x):a_{k-1}\le x \le a_k\}$.
And we define
$L(f,\pi)=\sum\limits_{k=1}^Km_k(f)(a_k-a_{k-1})$, where $m_k(f)=\inf\{f(x):a_{k-1}\le x \le a_k\}$.
Definition 1: We say that $f$ is Riemann-integrable on $[a,b]$ if and only if for every $\epsilon>0$ there exists a $\delta$ such that for every partition $\pi$ of $[a,b]$ with $||\pi||<\delta$, we have
$U(f,\pi)-L(f,\pi)<\epsilon.$
Definition 2: We say that $f$ is Riemann-integrable on $[a,b]$ if and only if
$\inf\{U(f,\pi):\pi\text{ is a partition of }[a,b]\}=\sup\{L(f,\pi):\pi\text{ is a partition of }[a,b]\}$.
What I want to prove is that the definitions are equivalent.
I think I can prove that defintiion 1 implies definition 2, the problem is that defintion 2 implies definition 1.
My attempt:
Assume definition 2. Let $\epsilon>0$ be given. There must exist a partition $\pi_1$ such that
$U(f,\pi_1)-\inf\{U(f,\pi):\pi\text{ is a partition of }[a,b]\}<\epsilon/2.$ There must also be a partion $\pi_2$ such that
$\sup\{L(f,\pi):\pi\text{ is a partition of }[a,b]\}-L(f,\pi_2)<\epsilon/2.$
Then we have
$U(f,\pi_1)-\epsilon/2<\inf\{U(f,\pi):\pi\text{ is a partition of }[a,b]\}=\sup\{L(f,\pi):\pi\text{ is a partition of }[a,b]\}<L(f,\pi_2)+\epsilon/2.$
Let $\pi_3$ be the partition of $[a,b]$ that contains all the points in the partitions $\pi_1$ and $\pi_2$. It is then easy to show that $U(f,\pi_1)\ge U(f,\pi_3)$ and $L(f,\pi_2)\le L(f,\pi_3).$
So we get:
$U(f,\pi_3)\le U(f,\pi_1)<L(f,\pi_2)+\epsilon\le L(f,\pi_3)+\epsilon.$
What I am now thinking of doing is choosing $\delta$ to be $\min_{1\le k\le K(\pi_3)}|a_k(\pi_3)-a_{k-1}(\pi_3)|$. But I am not able to prove that this will work. The problem is that even though a partition $\pi$ satisfying $||\pi||<\delta$ we dont know if it is a refinement of $\pi_3$ and if it is not a refinement I can't see that it will work. Do you see how to solve this? Is there a way to do it with this proof, or do we have to do something else?
