Choose an arbitrary partition P, and epsilon positive. Show that there is an equally-spaced partition E such that L(e^x,E)+epsilon > L(e^x,P).
Hello
I've been studying Darboux integrals, and I can't solve the following problem: Pn is the set of all equally spaced partitions of [0,1] such that Pn = {0, 1/n, 2/n, ... 1}
f(x) = e^x
I'm trying to prove that the integral from 0 to 1 of e^x = sup{L(f,Pn)}
In other words that the integral is equal to the supremum of the set of lower sums of f on equally spaced partitions.
From Darboux's definition of the integral, I know that the integral is equal to the supremum of the lower sums over ALL partitions. How can I limit that to just equally spaced partitions?