Proof or Counterexaple that a_n and b_n converge

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In summary, the conversation discusses a counterexample problem where it is stated that if a_n+b_n and a_n-b_n converge, then a_n and b_n must also converge. The person has tried various possibilities of a_n and b_n but could not find a counterexample. They mention the need to come up with something similar to the previous examples, where a_n+b_n and a_n-b_n converge, but a_n and b_n alone do not. The other person suggests trying to prove it and gives a hint to consider adding the two sequences for n>(N+M). The first person updates their question, stating that they were doing many similar counterexample problems and this one may not actually be a counterexample.
  • #1
Nana
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Homework Statement



If a_n+b_n and a_n-b_n converge then a_n and b_n must converge.
This MAY be a counterexample problem.

Homework Equations



N/A

The Attempt at a Solution



I have tried the following possibilities out of a_n and b_n and none of these give a counterexample:
Let a_n and b_n= (-1)^n , 1/n , 1/n^2 , (-1)^2n , sqrt(n)
I need to come up with something similar to this that holds true that a_n+b_n converge, a_n-b_n converge, but a_n and b_n alone do not converge.
 
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  • #2
Nana said:

Homework Statement



If a_n+b_n and a_n-b_n converge then a_n and b_n must converge.
This is going to be a counterexample problem.

Homework Equations



N/A

The Attempt at a Solution



I have tried the following possibilities out of a_n and b_n and none of these give a counterexample:
Let a_n and b_n= (-1)^n , 1/n , 1/n^2 , (-1)^2n , sqrt(n)
I need to come up with something similar to this that holds true that a_n+b_n converge, a_n-b_n converge, but a_n and b_n alone do not converge.

How do you know there is a counterexample?
 
  • #3
Why are you so sure there is a counterexample?
 
  • #4
LCKurtz said:
How do you know there is a counterexample?

We have been doing similar problems and they were all counterexamples, but there is a possibility that it is not.
 
  • #5
LCKurtz said:
How do you know there is a counterexample?

PeroK said:
Why are you so sure there is a counterexample?

It may not actually be. Thanks, I have updated my question. We were doing lots of counterexamples and it led me to believe that was one of them.
 
  • #6
You should try proving it. Because it is true.
 
  • #7
Hint: the two rows converge for n>N and for n>M respectively, what happens if you look at n>(N+M) and add the two sequences?
 

FAQ: Proof or Counterexaple that a_n and b_n converge

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

The definition of convergence for a sequence is that as the terms of the sequence approach infinity, the values of the terms get closer and closer to a single value. This value is known as the limit of the sequence. In other words, the sequence converges if and only if the terms eventually become arbitrarily close to some fixed value.

2. How can I prove that a sequence converges?

In order to prove that a sequence converges, you must show that the terms of the sequence approach a single value as n (the index of the sequence) approaches infinity. This can be done by using the epsilon-delta definition of a limit, which states that for any positive number ε, there exists a positive integer N such that for all n ≥ N, the terms of the sequence are within ε of the limit value.

3. What is a counterexample in regards to convergence of a sequence?

A counterexample is a specific example that disproves a statement or claim. In the case of a sequence, a counterexample would be a sequence that appears to converge, but upon further analysis, is found to not actually converge. This would disprove the statement that the sequence converges.

4. Can a sequence have multiple limits?

No, a sequence can only have one limit. This is because the definition of convergence requires that the terms of the sequence approach a single value as n approaches infinity. If a sequence had multiple limits, then the terms would not be approaching a single value, and therefore the sequence would not converge.

5. How does the rate of convergence affect the proof or counterexample?

The rate of convergence refers to how quickly the terms of a sequence approach the limit. A faster rate of convergence means that the terms of the sequence are getting closer to the limit value at a faster rate. This can affect the proof or counterexample in that a faster rate of convergence can make it easier to prove that a sequence converges, while a slower rate of convergence may require more complex techniques to prove convergence or find a counterexample.

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