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Uniform Convergence of a Sequence

Date: 16 Aug 1995 12:12:49 -0400
From: sbever
Subject: Uniform Convergence


I assert that uniform convergence of a sequence of functions
on a given interval to a funcion, f, is a "stronger" statement
than pointwise convergence of the sequence to f(x) for each x
in the interval since uniform convergence tells us that there
comes a point in the sequence after which all the functions in
the sequence stay within an infinitely small distance of f on
the entire interval, while pointwise convergence only gaurantees
that at each x, there comes point at which the sequence of
functions stay infinitely close to f at x, but there is
no guarantee that these same functions that are close to f at x
stay that close to f at another x.  We can find
examples where a sequence of continuous functions can converge
pointwise to a function which is not continuous, yet fail to
converge uniformly to that function (since it is discontinuous).


1. Is my thinking correct?

2. Can you state other reasons why uniform convergence is a
   "stronger" statement than pointwise convergence?

Date: 16 Aug 1995 12:39:39 -0400
From: Dr. Ethan
Subject: Re: Uniform Convergence

Yep, you got it - that is precisely the difference.  There really isn't much more 
to say except that uniform convergence implies pointwise convergence but 
not the other way around, as you already noticed.

Ethan- Doctor On Call
Associated Topics:
College Analysis

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