A question about Upper Darboux Integrals

[Artusartos]

In this link:

http://math.berkeley.edu/~scanez/courses/math104/fall11/homework/hw10-solns.pdf

For qustion 32.6, I'm not sure if I'm understanding how there can be a "sequence" of upper and lower darboux integrals.

So (for example), what is the difference between $$U_{10}$$ and $$U_{11}$$? Does it mean that $$U_{10}$$ is the upper darboux integral when there are 10 partitions (10 rectangles)...and $$U_{11}$$ is the upper darboux integral when there are 11 partitions? So, if we have $$U_1$$, does it mean that there is only one partition (so only one rectangle)?

Thanks in advance

 

[Vargo]

No, for each n, U_n is just any upper Darboux sum. It corresponds to some partition P_n which could have any number of subintervals. Likewise L_n is any old sequence of lower Darboux sums which corresponds to some other partition Q_n. The hypothesis is that the difference (U_n-L_n) converges to 0.

There is no stipulated relation between consecutive partitions in the sequence. In particular they are not assumed to be refinements of previous partitions.