# Power series,convergence of sequences of functions, and uniform limits

The question is in attached file.

Suppose a sequence of continuous functions, { ?n }, has the property that ?n  ? and

 > 0,   > 0 such that if | x - y | <  then n, | ?n (x) - ?n (y)| <  Prove that ? is

continuous.

https://brainmass.com/math/real-analysis/power-series-convergence-sequences-functions-uniform-limits-15066

#### Solution Preview

Please see the attachment.

Suppose a sequence of continuous functions, { ƒn }, has the property that ƒn ƒ and

> 0, > 0 such that if | x - y | < then n, | ƒn (x) - ƒn (y)| < Prove that ƒ is

continuous.

Proof. In order to show that ƒ is ...

#### Solution Summary

This is a proof regarding a sequence of continuous functions.

$2.19