Uniform Convergence of a Sequence

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

Discussion:

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).

Questions:

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