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Lim inf/sup innequality question

  1. Mar 10, 2009 #1
    [tex]
    \liminf _{n->\infty} x_n+\limsup _{n->\infty} y_n\leq \limsup _{n->\infty} (x_n+y_n)\leq\limsup _{n->\infty} x_n+\limsup _{n->\infty} y_n\\
    [/tex]

    proving the first part:
    [tex]
    \limsup _{n->\infty} (x_n+y_n)\leq\limsup _{n->\infty} x_n+\limsup _{n->\infty} y_n\\
    [/tex]

    lim sup is the supremum of all the limits of the subsequences

    this is true because of some law regarding the sum of two subsequences

    correct??
     
  2. jcsd
  3. Mar 10, 2009 #2
    why this is true
    x_n and y_n are bounded (as n->infinity)
    lim sup x_n >=lim x_r_n
    lim sup y_n >=lim y_r_n
    lim sup x_n+lim sup y_n>=lim x_r_n+lim y_r_n=lim(x_r_n+y_r_n)
    on what basis they get to lim(x_r_n+y_r_n) from sum of limits??

    and then
    lim(x_r_n+y_r_n)=limsup(x_n +y_n)
    how they got to this conclusion??
     
  4. Mar 11, 2009 #3
    whats the theory behind these operations
    ??
     
  5. Mar 11, 2009 #4

    Office_Shredder

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    Because (this has to be one of the first things you learned about limits), the sum of two limits is equal to the limit of the sum of their arguments (or whatever the thing inside the limit is called). It's a pretty straightforward proof that you should try to do if you haven't seen it.
    Considering you haven't told us what x_r_n is supposed to be, it's impossible to answer this
     
  6. Mar 11, 2009 #5
    x_r_n is a subsequence of x_n
    y_r_n is a subsequence of y_n

    lim(x_r_n+y_r_n)=limsup(x_n +y_n)
    how they got to this conclusion??
     
    Last edited: Mar 11, 2009
  7. Mar 11, 2009 #6
    how to understand this last part?
     
  8. Mar 11, 2009 #7
    in the solution they say that
    because (x_r_n +y_r_n) is convergent then it equals limsup(x_n +y_n)

    so what that it is a convergent?
     
  9. Mar 11, 2009 #8
    why its true
     
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