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Math Help - Sequences of Functions

  1. #1
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    Sequences of Functions

    I have been studying pointwise and uniform convergence of sequences of functions and i am stuck with this question,

    Show that lim(Arctan nx) = (pi/2)sgnx for x in R.

    I believe that we will have to take three cases for the limit function f(x), namely when it is equal to -(pi/2), 0 and (pi/2).
    But i cannot figure out , a way to show that |fn(x) -f(x)|< epsilon.


    Please HELP.
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  2. #2
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    Re: Sequences of Functions

    Suppose you know that \arctan(x)\to\pi/2 as x\to\infty. That is, there is a function \mu mapping neighborhoods of  \pi/2 to neighborhoods of infinity: for every \varepsilon>0 and every x>\mu(\varepsilon) we have |\arctan(x)-\pi/2|<\varepsilon.

    Now fix an arbitrary x > 0. Let N(\varepsilon)=\lceil\mu(\varepsilon)/x\rceil where \lceil\cdot\rceil is the ceiling function. The essential property of N(\varepsilon) is that n>N(\varepsilon) implies nx>\mu(\varepsilon). Therefore, for any \varepsilon>0 and n>N(\varepsilon) we have |\arctan(nx)-\pi/2|<\varepsilon.

    This (and a similar argument for x < 0) proves that \arctan(nx) converges to (\pi/2)\mathop{\text{sgn}}(x) pointwise. Note that it does not converge uniformly because \sup_{x\in\mathbb{R}} |\arctan(nx)-\pi/2|= 1 for all n.
    Thanks from mrmaaza123
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