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Taking the lim operation on both sides

  1. Mar 16, 2009 #1
    is it legal to take the limit on both sides of an expression

    i particular

    when i have

    sup x_n <=sup y_n

    is it ok to do

    limsup x_n <=limsup y_n

    is the limit of the supremums group of subsequences is the limsup on the sequence
    ??
     
  2. jcsd
  3. Mar 16, 2009 #2
    One of the properties of the limit is that if [tex]a_n\le b_n[/tex] for all n, then [tex]\lim_{n\to\infty}a_n\le\lim_{n\to\infty}b_n[/tex] provided that these limits exist.
    Does this answer your question?
     
  4. Mar 17, 2009 #3

    Office_Shredder

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    This isn't quite that case yyat.

    Counterexample: xn = 1 if n=1, 0 otherwise
    yn = 1/2 for all n

    supxn = 1
    supyn = 1/2

    limsup xn = 0
    limsup yn = 1/2
     
  5. Mar 17, 2009 #4
    I don't see how this contradicts anything I wrote.

    But you are right, if transgalactic means that sup x_n=sup{x_n,n>=0}, then the statement is wrong by the counterexample you gave (with x_n, y_n switched). I assumed it meant sup{x_n,n>m}<=sup{y_n,n>m} for all m.
     
    Last edited: Mar 17, 2009
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