Help with proving a sequence converges

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Please do not post duplicate questions."In summary, the conversation is about proving that if the limit of sequence sn is L and the limit of the absolute value of the difference between sn and tn is 0, then the limit of sequence tn is also L. The conversation also includes a discussion on how to approach the proof and the use of an N-epsilon proof.
  • #1
jarvegg
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I am really stuck on this on

Let sn and tn be sequences. Suppose that lim sn = L (where s is a real number) and lim |sn - tn| = 0. Prove that lim tn = L.

I think this is going in the right way but i am not sure. If the lim an = A, then lim |an - A| = 0.

any help would be very nice.
 
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  • #2
Help proving a sequesnce converges

I am really stuck on this on

Let sn and tn be sequences. Suppose that lim sn = L (where s is a real number) and lim |sn - tn| = 0. Prove that lim tn = L.

I think this is going in the right way but i am not sure. If the lim an = A, then lim |an - A| = 0.

any help would be very nice.
 
  • #3
let N be so large that
|s_n|<epsilon/2
|s_n-t_n|<epsilon
what can you say about
|t_n|
?
hint
|A-B|<=|A|+|B|
 
  • #4
I know how to do a N Epsilon proof but don't you have to have |s_n + t_n - (s + t)| < epsilon??

And that is to show that the lim (s_n + t_n) = s + t. I need to know lim |s_n - t_n| = 0. i guess i am unsure how the abs effect the proof.
 
  • #6
I have merged the two threads.
 

FAQ: Help with proving a sequence converges

1. What is the definition of convergence for a sequence?

The definition of convergence for a sequence is that a sequence {an} converges to a limit L if for any positive number ε, there exists a positive integer N such that for all n > N, the distance between the terms an and L is less than ε. In other words, as n grows larger, the terms of the sequence approach the limit L.

2. How do I prove that a sequence converges to a specific limit?

To prove that a sequence converges to a specific limit, you can use the definition of convergence and show that for any ε > 0, there exists an N such that for all n > N, the distance between the terms of the sequence and the limit is less than ε. This can be done through various methods such as the squeeze theorem, the monotone convergence theorem, or using the definition of limits.

3. Can a sequence converge to more than one limit?

No, a sequence can only converge to one limit. This is because the definition of convergence states that the terms of the sequence must get closer and closer to the limit as n grows larger. If a sequence were to converge to more than one limit, this would mean that the terms get closer to multiple values, which is not possible.

4. How can I determine if a sequence diverges instead of converges?

To determine if a sequence diverges instead of converges, you can use the definition of convergence and show that for any limit L, there exists an ε > 0 such that for all N, there exists an n > N where the distance between the terms of the sequence and the limit is greater than ε. This means that the terms of the sequence do not approach a specific limit and therefore, the sequence diverges.

5. Are there any special cases when proving convergence for a sequence?

Yes, there are some special cases when proving convergence for a sequence. These include dealing with infinite or oscillating sequences, sequences with alternating signs, and sequences with recursive formulas. It is important to carefully analyze and understand the properties of these types of sequences in order to prove their convergence.

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