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

$U(f,P)-L(f,P)<0.001$

$\lambda P=max(x_k-x_k-1)$

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.