- #1

lys04

- 70

- 3

Was reading the Reimann integrals chapter of Understanding Analysis by Stephen Abbott and got stuck on exercise 7.2.5. In the solutions they went from having |f-f_n|<epsilon/3(a-b) to having |M_k-N_k|<epsilon/3(a-b), but I’m confused how did they do this. We know that fn uniformly converges to f, that means for any epsilon greater than 0 I can find natural number M st when n is greater than M then the distance from fn to f is less than epsilon for all x in the interval [a,b], but the supremums of fn and f in a sub interval [x_k-1,x_k] might occur at different values and I’m not sure how they’re meant to be at most epsilon apart.