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bill1158 science forum beginner
Joined: 09 Jul 2006
Posts: 12

Posted: Wed Jul 19, 2006 10:59 am Post subject:
uni. convergence



Let g : R > R be a continuous function such that g(1) = 0. If we
define
f_n(x) = g(x) x^n. how can we show that the sequence {fn} converges
uniformly to some function f on the set E = [0, 1].
My problem here is that there are two functions, f and g. 

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G.E. Ivey science forum Guru
Joined: 29 Apr 2005
Posts: 308

Posted: Wed Jul 19, 2006 11:44 am Post subject:
Re: uni. convergence



Quote:  Let g : R > R be a continuous function such that
g(1) = 0. If we
define
f_n(x) = g(x) x^n. how can we show that the sequence
{fn} converges
uniformly to some function f on the set E = [0, 1].
My problem here is that there are two functions, f
and g.
Actually, you have an infinite number of functions g, f_1, f_2, etc.! 
Suppose x<1. What is the limit of g(x)x^n as x goes to infinity? (Remember that g(x) is a constant with respect to n.) What is that limit if x= 1? To show that the convergence is uniform, given any epsilon you will have to find an N such that g(x)x^n< epsilon for ALL x between 0 and 1. That should be not be difficult for all x< 1 and trivial to show that it also works for x=1. 

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The World Wide Wade science forum Guru
Joined: 24 Mar 2005
Posts: 790

Posted: Wed Jul 19, 2006 5:11 pm Post subject:
Re: uni. convergence



In article
<1153306753.880150.11010@i42g2000cwa.googlegroups.com>,
"bill" <bilgiaslankurt@gmail.com> wrote:
Quote:  Let g : R > R be a continuous function such that g(1) = 0. If we
define
f_n(x) = g(x) x^n. how can we show that the sequence {fn} converges
uniformly to some function f on the set E = [0, 1].
My problem here is that there are two functions, f and g.

g is small for x in (1d, 1]. x^n > 0 uniformly on [0, 1d].
Put the two ideas together. 

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Lynn Kurtz science forum Guru
Joined: 02 May 2005
Posts: 603

Posted: Wed Jul 19, 2006 5:41 pm Post subject:
Re: uni. convergence



On 19 Jul 2006 03:59:13 0700, "bill" <bilgiaslankurt@gmail.com>
wrote:
Quote:  Let g : R > R be a continuous function such that g(1) = 0. If we
define
f_n(x) = g(x) x^n. how can we show that the sequence {fn} converges
uniformly to some function f on the set E = [0, 1].
My problem here is that there are two functions, f and g.

Here's a hint:
Look at two parts. You can make g(x)x^n small for x >= a which is
near, but less than, 1 because g is continuous, g(1) = 0, and x^n < 1.
Then for the rest of the interval [0,a] you can make g(x)x^n small
because g is bounded and x^n < a^n. Fill in the details.
Lynn 

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