Limits Proof

  • Thread starter khdani
  • Start date
  • #1
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Hello,
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 ?
 

Answers and Replies

  • #2
1,357
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What if b = 0?
 
  • #3
55
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if b=0 than a is negative and probably an < bn
so it still holds, but how do i prove that?
 
  • #4
1,357
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Consider the sequence an - bn. What can you say about this sequence?
 
  • #5
55
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can I say that
lim(an-bn) = (a-b) < 0
hence an < bn ?
 
  • #6
1,357
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You can, but it doesn't convince me that an < bn. Also, I think that what you're trying to prove is false.
 
  • #7
HallsofIvy
Science Advisor
Homework Helper
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Hello,
Please help me prove the following:
given lim(an)=a and lim(bn)=b if a<b prove that an < bn.
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.

can i say that if a/b < 1 than an<bn ?
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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