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A question about property of liminf and limsup

  1. Jun 4, 2010 #1
    If [tex]x_n\geq 0, y_n\geq 0[/tex] and [tex]\lim \limits_{n \to \infty }x_n[/tex] exists, we have [tex]\limsup\limits_{n\to\infty}(x_n\cdot y_n)=(\lim\limits_{n\to\infty}x_n)\cdot(\limsup\limits_{n\to\infty}y_n)[/tex]. But if [tex]\lim\limits_{n\to\infty}x_n<0[/tex], do we have analog equation(I guess [tex]\limsup\limits_{n\to\infty}(x_n\cdot y_n)=(\lim\limits_{n\to\infty}x_n)\cdot(\liminf\limits_{n\to\infty }y_n)[/tex])? and what change should be made to conditions to achieve the analog equation? Formal source of reference such as textbooks or webpages is recommended. Thanks!
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  3. Jun 6, 2010 #2

    Gib Z

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    Homework Helper

    Do you know how to prove the first result? Make a proof, take a careful look to see where you used the assumption that x_n is positive, and then it should be clear how things become changed when you assume x_n is negative.
  4. Jun 6, 2010 #3


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    As always with liminf and limsup, use the sign trick!
    [tex]\liminf (-a_j)=-\limsup(a_j).[/tex]
    If x_n<0, then -x_n>0, so we can apply your result (in the form [itex]\liminf(a_jb_j)=\lim(a_j)\liminf(b_j)[/itex] for a_j,b_j nonnegative and a_j convregent, i.e. sup replaced by inf) to get:

    [tex]\limsup (x_ny_n)=-\liminf (-x_ny_n)=-\lim(-x_n)\liminf(y_n)=\lim(x_n)\liminf(y_n)[/tex]
  5. Jun 7, 2010 #4
    Thank you Landau, it's really a good idea!
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