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What does "norms less than $\delta$" refers to in the context of Riemann sum?

Given this problem "Prove if $\int_{a}^{b} f(x)$ exists, then for every $\epsilon>0$, there is a $\delta >0 $ such that $|\sigma_1 -\sigma_2|<\epsilon$, if $\sigma_1$ and $\sigma_2$ are Riemann sums of $f$ over partitions P$_1$ and P$_2$ of $[a,b] $ with norms less than $\delta$."

I am trying to understand this problem that I posted Prove that if $\int_{a}^{b} f(x)$ exists, $\delta >0 $ such that $|\sigma_1 -\sigma_2|<\epsilon$ ), I did receive one answer but I am not understanding the how and the why of his argumentation leads to the conclusion that "the norms is less than $\delta$" (sub-question "norms" why plural?). What should be understood from by "the norms is less than $\delta$"?

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The plural concerns the two subdivisions or partitions $P_1$ and $P_2$.

if $P=(x_0,x_1,x_2,...,x_n) $ is a subdivision of $[a,b] $ then what you call norm of $P $ is

$$\|P\|=\max_{0\le i\le n-1}\{x_{i+1}-x_i\} $$

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Before this there should be a definition of the "norm of a partition". The norms being referred to here are the norm of $P_1$ and the norm of $P_2$; both are less than $\delta $.

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  • $\begingroup$ I would be happy to address any issues with my answer. If someone would prefer not to comment publicly, I can be reached privately as well (instructions are in my profile). $\endgroup$ – Mark S. Jul 26 '17 at 14:46

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