Explore BrainMass

# Uniformly Cauchy Sequence of Continuous Functions

Not what you're looking for? Search our solutions OR ask your own Custom question.

This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!

Let f_n : [0,1] -> R be a sequence of continuous functions such that for each n in N (natural numbers), f_n is differentiable on (0,1). Suppose that f_n(0) converges to some number, denoted f(0), and also suppose that the sequence (f'_n) converges uniformly on (0,1) to some function g: (0,1) -> R. Prove that the sequence (f_n) converges uniformly. What is its limit?

https://brainmass.com/math/graphs-and-functions/uniformly-cauchy-sequence-of-continuous-functions-39330

#### Solution Preview

Please see the attached file for the complete solution.

Problem:

Let f_n : [0,1] -> R be a sequence of continuous functions such that for each n in N ( natural numbers), f_n is differentiable on (0,1). Suppose that f_n(0) converges to some number, denoted f(0), and also suppose that the sequence (f'_n) converges uniformly on (0,1) to some function g: (0,1) -> R. Prove that the sequence (f_n) converges uniformly. What is its limit? I was ...

#### Solution Summary

Uniformly cauchy sequence of continuous functions are investigated and discussed The solution is detailed and well presented.

\$2.49