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Thread: upper bound turning into supremum

  1. #1
    MHF Contributor
    Nov 2008

    upper bound turning into supremum

    i proved that sin (1/x)<1/x

    prove that sup{xsin (1/x)|x>0}=1

    if we say that A={xsin (1/x)|x>0}
    xsin (1/x)<x(1/x)=1

    so one is upper bound

    now i need to prove that there is no smaller upper bound so that 1 is the supremum

    suppose that "t" is our smaller upper bound t<1 and epsilon=1-t
    now i need to do some limit definition and |f(x)-1|<epsilon

    from that i need to get that t>1 so 1 is the only supremum
    how to do that
    Last edited by transgalactic; Nov 16th 2010 at 11:36 PM.
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  2. #2
    Senior Member Shanks's Avatar
    Nov 2009
    $\displaystyle \frac{\sin t}{t}\to 1$ when $\displaystyle t>0,t\to 0$.
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