Proof or Counterexaple that a_n and b_n converge

[Nana]

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.

 

[LCKurtz]

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?

 

[PeroK]

Why are you so sure there is a counterexample?

 

[Nana]

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.

 

[Nana]

How do you know there is a counterexample?

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.

 

[kduna]

You should try proving it. Because it is true.

 

[dirk_mec1]

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?