So here is the question, that I am just really stuck on :

Problem. Let x(n) = (-1)^n / n, and A be a real number. Prove the

following:

there exists such a 1-to-1 mapping (bijection) p : N --> N that

an infinite series with a generic term x(p(n)) converges and the sum of

this series is equal to A .

ok, So I understand that as n approaches infiniti, this series is really getting to zero from both sides. understood. so we need to prove it's an injection and a surjection. for an injection, do you just have to prove the basic fact that if f(x_1) = f(x_2), then x_1=x_2 ?? Now with the surjection, I am just confused in general, and don't even know how to approach it.

Now with this term A, we want to prove that the series converges (so would i just show the limit, as stated above? or how would i do this with a real analysis approach??) and also, the sum of this series is what? how can you conclude that its sum is A??

Thank you for the help. Just wanted to make clear what I understand and don't understand, etc.