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Math Help - Question about uniform convergence

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    Question about uniform convergence

    Hi, so Im studying for finals and am stuck on a problem. I got referred to this forum from a friend, and I had a quick question about uniform convergence.
    we have a function which is:

    f_n(x)\frac{nx}{nx+1} for all n greater than or equal to 1.

    Is this uniformly convergent at [0,1]?
    Then if we were to fix r>0, is it uniformly convergent at [r, infinity]?

    Also, there was another part of the problem. the professor wrote something like
    f(x) = \lim_{x\to\infty}f(x^n)
    He's a messy writer, and i do believe that was what he wrote, but Im not too sure if that fits into the problem at all, or if it even does.

    Any help is appreciated. Thanks!
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Quote Originally Posted by wontonsoup View Post
    f_n(x)\frac{nx}{nx+1} for all n greater than or equal to 1. Is this uniformly convergent at [0,1]?
    Every f_n is continuous on [0,1] , however:

    f(x)=\displaystyle\lim_{n \to \infty}f_n(x)=\begin{Bmatrix}0&\textrm{if}&x=0\\1&  \textrm{if}&x\in(0,1]\end{matrix}

    is not continuous on [0,1], so by a well known theorem, the convergence is not uniform.

    Then if we were to fix r>0, is it uniformly convergent at [r, infinity]?
    Yes, now

    f(x)=\displaystyle\lim_{n \to \infty}f_n(x)=1

    Prove that for every \epsilon>0 there exists n_0\in\mathbb{N} such that

    |f_n(x)-f(x)|<\epsilon for n\geq n_0 and for all x\in[r,+\infty)

    Also, there was another part of the problem. the professor wrote something like
    f(x) = \lim_{x\to\infty}f(x^n)
    He's a messy writer, and i do believe that was what he wrote, but Im not too sure if that fits into the problem at all, or if it even does.
    Possibly is:

    f(x)=\displaystyle\lim_{n \to \infty}f_n(x)

    Fernando Revilla
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