A sequence of functions fn defined on a set E is said to be equicontinuous on E if for every ?

Name: A sequence of functions fn defined on a set E is said to be equicontinuous on E if for every ? > 0, there exists a ? > 0 such that |fn(x) - fn(y)| < ? whenever |x - y| < ?, x ? E, y ? E and n ? N. Prove that, if an equicontinuous sequence of functions fn converges pointwise to f on a set E, then f is uniformly continuous on E.
Uploaded: 2023-06-05T23:24:02-08:00
Duration: 3 min 43 s
Channel: Adi S
Description: A sequence of functions fn defined on a set E is said to be equicontinuous on E if for every ? > 0, there exists a ? > 0 such that |fn(x) - fn(y)| < ? whenever |x - y| < ?, x ? E, y ? E and n ? N. Prove that, if an equicontinuous sequence of functions fn converges pointwise to f on a set E, then f is uniformly continuous on E.

Question

A sequence of functions fn defined on a set E is said to be equicontinuous on E if for every ? &gt;

Vikash Ranjan · Accepted Answer

1. We are given that the sequence of functions {f_n} is equicontinuous on E. This means that for

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