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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.

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(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.

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