Convergence of a sequence, {(-1)^n}n>=1

[missavvy]

Homework Statement

Prove that {(-1)ⁿ} n>=1 does not converge

Homework Equations

The Attempt at a Solution

If i define two subsequences, say {(-1)²ⁿ} = A and {(-1)²ⁿ⁺¹} =B of that original sequence, then
A converges to 1 and B converges to -1 ?

Is this correct at all?

 

[Mark44]

Homework Statement

Prove that {(-1)ⁿ} n>=1 does not converge

Homework Equations

The Attempt at a Solution

If i define two subsequences, say {(-1)²ⁿ} = A and {(-1)²ⁿ⁺¹} =B of that original sequence, then
A converges to 1 and B converges to -1 ?

Is this correct at all?

The sequence {(-1)²ⁿ} converges to 1, and the other sequence converges to -1, but A and B aren't sequences (they're numbers), so you shouldn't talk about them converging.

What about assuming that {(-1)ⁿ}, for n>=1, converges, and arriving at a contradiction?

 

[missavvy]

Hm okay, so should I use the cauchy definition of convergence? (If yes, how would I put the "limit" in?)

 

[micromass]

In your OP you showed that you have two subsequences that converge to different limits. This indeed shows that our series cannot be convergent. To me, it's a valid proof...

Am I making a mistake here??