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Limits Proof

  1. Dec 10, 2008 #1
    Please help me prove the following:
    given lim(an)=a and lim(bn)=b if a<b prove that an < bn.

    can i say that if a/b < 1 than an<bn ?
  2. jcsd
  3. Dec 10, 2008 #2
    What if b = 0?
  4. Dec 10, 2008 #3
    if b=0 than a is negative and probably an < bn
    so it still holds, but how do i prove that?
  5. Dec 10, 2008 #4
    Consider the sequence an - bn. What can you say about this sequence?
  6. Dec 10, 2008 #5
    can I say that
    lim(an-bn) = (a-b) < 0
    hence an < bn ?
  7. Dec 10, 2008 #6
    You can, but it doesn't convince me that an < bn. Also, I think that what you're trying to prove is false.
  8. Dec 10, 2008 #7


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    You can't prove it- it isn't true. What you can prove is that for n large enough, an< bn. Use the definition of limit with [itex]\epsilon[/itex] less that half the difference between a and b.

    But you cannot say anything about an and bn for smaller values of n.

    No, that's not true either. Again, it is only true for "sufficiently large" n.

    Got example, an= 1/n converges to 1 while bn= 1/2n for n= 1 to 1000000, bn= 2- 1/n for n> 1000000 converges to 2 (so a= 0< 2= b and a/b= 1/2< 1) but an< bn only for n> 1000000. And you should be able to see how to make examples where that is true only for n> whatever number you want.
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