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Thread: Real Analysis - Sequences

  1. #1
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    Real Analysis - Sequences

    Hey, I'm pretty sure I already solved part a. of this question, but decided to put it up just in case. Any help would be greatly appreciated. Thanks.

    Let fn(x) = nx / (1 + nx^2)

    a. Find the pointwise limit of (fn) for all x Є (0, infinity).
    b. Is the convergence uniform on (0, infinity)?
    c. Is the convergence uniform on (0,1)?
    d. Is the convergence uniform on (1, infinity)?
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  2. #2
    Member TheMasterMind's Avatar
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    Quote Originally Posted by ajj86 View Post
    Hey, I'm pretty sure I already solved part a. of this question, but decided to put it up just in case. Any help would be greatly appreciated. Thanks.

    Let fn(x) = nx / (1 + nx^2)

    a. Find the pointwise limit of (fn) for all x Є (0, infinity).
    b. Is the convergence uniform on (0, infinity)?
    c. Is the convergence uniform on (0,1)?
    d. Is the convergence uniform on (1, infinity)?
    Note that for x≥1, f_n(x) = nx/(1+nx^2) < nx/(nx^2) = 1/x ≤ 1 so each of the $\displaystyle f_n$ is bounded above by $\displaystyle 1$ on $\displaystyle [1, infinity)$ so since $\displaystyle f_n$ is continuous on $\displaystyle [0, 1]$ and $\displaystyle [0, 1]$ is compact we have by the extreme value theorem that $\displaystyle f_n(x)$ is bounded above on $\displaystyle [0, 1]$ taking the greater of $\displaystyle 1$ and the upper bound for $\displaystyle f_n(x)$ on $\displaystyle [0, 1]$, upper bound for $\displaystyle f_n(x)$ on $\displaystyle [0, infinity]$. Therefore each one of the $\displaystyle f_n$ is bounded above on $\displaystyle [0, infinity]$.

    to find the pointwise limit of the $\displaystyle f_n$, we see that for $\displaystyle x>0$

    $\displaystyle f_n(x)$

    $\displaystyle = \frac{nx}{1+nx^2}$

    $\displaystyle = \frac{1}{(1/(nx) + x)}$

    this approaches 1/x as n\forwardarrow\infinity now for every n, $\displaystyle f_n(0) = 0$, we have

    n---->infinity lim f_n(x) = {0 if x=0, 1/x if x≠0}

    this shows it is not bounded above on $\displaystyle [0, infinity]$. It follows that the $\displaystyle f_n$ cannot converge uniformly
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