# Real Analysis : Cauchy Criterion for Uniform Convergence

Prove that A sequence of functions (f_n) defined on a set A subset or equal to R converges uniformly on A if and only if for every e>0(epsilon) there exists an N belong to N such that Absolute value of f_n (x)-f_m (x)<e for all m,n>=N and all x belong to A.

Â© BrainMass Inc. brainmass.com February 24, 2021, 2:34 pm ad1c9bdddfhttps://brainmass.com/math/real-analysis/real-analysis-cauchy-criterion-uniform-convergence-30034

#### Solution Preview

Please see the attached file for the complete solution.

Thanks for using BrainMass.

(Cauchy criterion for uniform convergence).prove that A sequence of functions (f_n) defined on converges uniformly on A if and only if for every there exists an N belong to N such that for all m,n>=N and all .

Definition (Uniform Convergence). The sequence converges uniformly to f(x) on the set A if for every , ...

#### Solution Summary

The Cauchy criterion for uniform convergence is investigated. The solution is detailed and well presented.