Let Sum of an(sign of sum) be given.For each n belong to N let p_n=an if a_n is positive and assign p_n=0 if a_n is negative.In a similar manner,let q_n=a_n if an is negative and q_n=0 if a_n is positive.
1-Argue that if Sum a_n diverges then at least one of sum p_n or sum q_n diverges.
2- show that if sum a_n converges conditionally then both sum p_n and sum q_n diverge.
First we want to express a_n in terms of p_n and q_n:
If a_n > 0 , then a_n = p_n, and q_n = 0;
if a_n < 0 , then a_n = - q_n, and p_n = 0;
therefore a_n = p_n - q_n and Sum a_n = Sum p_n - Sum q_n.
1. Prove by contradiction:
Otherwise if both sum p_n and sum q_n converges( say P and Q, ...
This is a proof regarding the convergence and divergence of a sum.