[Math] How to show these two definitions of the Riemann integral are equivalent

Question

Let $f:[a,b]\longrightarrow \mathbb{R}$

Let $P=\{a=t_0<t_1<\cdots<t_n=b\}$ be partition of $[a,b]$

$P^*=(P,\xi)$ , $\xi=(\xi_1,\xi_2,\cdots,\xi_n)$ , $t_{i-1}\le\xi_i \le t_i$

We define $\displaystyle \sum (f,P^*)=\sum_{i=1}^{n} f(\xi_i) \cdot (t_i-t_{i-1}) $

$||P||=\underset {1\le i\le n}{\text{max}} (t_i-t_{i-1})$

---

Definition $1$

We define $\displaystyle \int_a^b f =\lim_{||P||\to0} \sum(f,P^*)$ if the limit exists

$\forall \epsilon >0 , \exists\delta>0: \forall P^* ,||P||<\delta \Longrightarrow \left|\displaystyle \int_a^b f-\sum(f,P^*)\right|<\epsilon$

---

Definition $2$

Another definition of Riemann integral:

If there is a number $L$ such that:

$\forall \epsilon >0 , \exists P_{\epsilon} $ partition of $[a,b]: \forall P^*=(P,\xi) , P_{\epsilon}\subset P $ finer partition $ \Longrightarrow |L-\sum(f,P^*)|<\epsilon$

then $\displaystyle \int_a^b f=L$

---

We note that Definition $1$ implies Definition $2$, but what about the converse ?

Any hints would be appreciated.

Answer 1

This is a sketch presented in Apostol's Mathematical Analysis. Can you fill in the details? There aren't too many left! You need to look at the $S_1,S_2$ carefully.

Let $\int_a^b f(x)dx =I$, $M=\sup\{|f(x)|:x\in[a,b]\}$. Given $\epsilon >0$; choose $P_\epsilon$ such that $U(P_\epsilon,f)<I+\dfrac \epsilon 2$. [Here $U$ is the upper Darboux/Riemann sum of $f$] Let $N$ be the number of points of division in $P_\epsilon$ and let $\delta=\dfrac{\epsilon}{2MN}$. If $\Vert P\Vert <\delta$ put $$U(f,P)=\sum M_kf(x) \Delta_x=S_1+S_2$$ where $S_1$ is the sum over the terms that belong to the subintervals of $P$ that have no points of $P_\epsilon$ and $S_2$ is the remianing sum. Then $$S_1\leq U(P_\epsilon,f)<I+\frac \epsilon 2$$ $$S_2\leq NM\Vert P\Vert <NM\delta=\frac\epsilon 2$$ thus $$U(P,f)<I+\epsilon$$ Analogously, $$L(f,P)>I-\epsilon$$ if $\Vert P\Vert<\delta^\prime$, for a suitable $\delta^\prime$. Thus $$|S(P,f)-I|<\epsilon$$ if $\Vert P\Vert<\min\{\delta,\delta^{\prime}\}$.

NOTATION $U(f,P)$ denotes the upper sum of $f$ w.r.t $P$, that is $$\sum M_k \Delta x_k$$ where $M_i$ is the supremum of $f$ over the $i$-th subinterval of $P$. Analogously, $L(f,P)$ is the lower sum of $f$ w.r.t. $P$. Finally $S(f,P)$ is any Riemann sum of a tagged partition $P$.