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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.

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  • $\begingroup$ Hello - this could very well be a definition of Riemann integrability. So to get a good answer, you should state the definition of Riemann integrability that you are using, and also explain your own understanding of this property. $\endgroup$ – Hans Engler Jun 30 '14 at 18:32
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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.

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Hint: If $\mathcal{P}$ is a partition such that $U(f,\mathcal{P}) - L(f,\mathcal{P}) < \epsilon$ then you know $$U(f,\mathcal{P}) - \int f + \int f - L(f,\mathcal{P}) < \epsilon$$ as well as $$U(f,\mathcal{P}) - \int f \geq 0 \quad \text{and}\quad \int f - L(f,\mathcal{P}) \geq 0$$ Try to conclude why this partition $\mathcal{P}$ will work for your cause. Here $U(f,\mathcal{P})$ and $L(f,\mathcal{P})$ are the upper and lower sums respectively.

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Think about definitions of what upper and lower integrals are. Specifically, they are infimum and supremum of integrals of step functions. Use the supremum and infimum properties (i.e. if A is a supremum of set S, then for any $\epsilon>0$ there exists $a$ in S such that $A-\epsilon<a\le A$)

Then you can get step functions $\psi_+$ and $\psi_-$ such that $\psi_-\le f\le \psi_+$ on $[a,b]$. Suppose that $\psi$ is a step function on the partition $(x_0,\dots,x_n)$ such that $I(\psi)$ equals given Riemann sum and so $\psi_- \le \psi \le \psi_+$. Then what can you say about $I(\psi)$ in comparison with upper and lower integral, using sup/inf properties?


edit: I just realized that my definitions in terms of step functions might not make sense to you (there are different definitions I guess).. if so, sorry about that.

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