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

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

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