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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.
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.
science forum Guru

Joined: 24 Mar 2005
Posts: 790

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

In article
"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 (1-d, 1]. x^n -> 0 uniformly on [0, 1-d].
Put the two ideas together.
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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