Solution of the 6 th Homework – On progress –

1 Exercise 1 The series ∑∞ n m an converges, by de nition, exactly when the partial sum SN ∑N n m an converges as N →∞. Now using the completeness of R, this happens exactly when (SN )∞ N m is a Cauchy sequence. So it su ces to show that this is equivalent to the condition given in the exercise. This is almost trivial, but we introduce the proof anyway to please some meticulous readers. • Suppose that the condition in the exercise holds. Let ε > 0 be arbitrary and let N be as in the condition. Then whenever p , q ≥ N , either p ≥ q or p ≤ q. In the former case, |Sp − Sq | p ∑