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
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