Exercise $\bf 5.1.7$: Suppose $f:[a,b]\to\Bbb R$ is Riemann integrable. Let $\epsilon\gt0$ be given. Then show that there exists a partition $P=\{x_0,x_1,\ldots,x_n\}$ such that if we pick any set of numbers $\{c_1,c_2,\ldots,c_n\}$ with $c_k\in[x_{k-1},x_k]$ for all $k$, then $$\left|\int_a^b f-\sum_{k=1}^n f(x_k)\Delta x_k\right|\lt\epsilon.$$
I'm trying to do some example problems in analysis, as I'm struggling to make some mental connections with some of our topics. Anybody able to give me some hints or even an example proof of this problem? This seems to be a problem attempting to prove the validity of riemann sums. Listed below is the definition of riemann integrable that I am working with.
A function is Riemann Integrable if it's lower and upper Darboux integral are equal to each other.
Best Answer
The definitions of Riemann integral a la Darboux and as a limit are equivalent. You ask for a proof in one direction.
The proof is typical and uses the simple fact:
if $\quad \alpha\le c \le \beta \quad$ and $\quad \alpha\le d \le \beta \quad$ then $\quad |c-d| \le \beta-\alpha \;\;$.
So, if $P$ is a partition of $[a,b]$, however tagged, from the theory, following the Darboux definition, one has $$\left|\int_a^b f(x)\,dx-R(f,P) \right|\le U(f,P)-L(f,P)$$
Let $\varepsilon>0$. Then the (Darboux) integrability criterion assures the existence of a partition $P$ of $[a,b]$ such that $$U(f,P)-L(f,P)<\varepsilon$$ Combine everything to get the result.