I simply don't know how to do that. I'll let the norm of the partition be less than delta=epsilon, but when working with the Riemann sum, and when t_i = x_i, I'll be left with f(x_i)(x_i - x_i-1) < f(x_i)*epsilon.
Since I don't know what f(x_i) is, I can't put a bound on it. I don't know...
Homework Statement
Suppose that f:[a, b] → ℝ is a function that is zero for all x ∈ [a, b] except for the values x_1,x_2,…,x_k. Find ∫[a b](f(x)dx) and prove your result.
Homework Equations
Definition of a Riemann integrable function...
Ah, I proved in an earlier example that if 0 < r < 1, then nr^n converges to zero.
I realize now that the last pair of terms should be |An+1 -An|. The inequality should be true now: |Am - Am-1| + |Am-1 - Am-2| +...+ |An+1 -An|< n*r^n. So, am I done since I've shown that |An+1 - An| < ... <...
Homework Statement
Let 0 < r < 1. Let {A_n} be a sequence of real numbers such that |A_n+1 - A_n| < r^n for all naturals n. Prove {A_n} converges.
Homework Equations
A sequence of real numbers is called Cauchy, if for every positive real number epsilon, there is a positive integer N...