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Integral
Hi, I have a question, but I am not sure about this.
So we have the function $f(x)=1/x$ at the interval [1,2]. I have to find $0<\delta$ such that for every partition with $\lambda P<\delta$ such that
$U(f,P)L(f,P)<0.001$
$\lambda P=max(x_kx_k1)$
So what I did is to take partition with equal length, so that at the end of the calculation I would get something like 1/0.001<n and I could say something about the length $\lambda P$
(I decided to upload a picture because it's hard writing here math)
The solution that I wrote feels like a more specific case rather than general. I showed it for every partition with equal length although it is true for any partition. How can I solve this without having to chose partition with equal length?
Thank you.

Re: Integral
For a random partition $\displaystyle P$, the series $\displaystyle \sum_k f(x_{k1})f(x_k)$ is telescopic, so $\displaystyle U(f,P)L(f,P)$ equals a constant times $\displaystyle \frac{1}{n}$.