Problem Statement: Suppose that converges absolutely. Prove that converges for every .
Proof: We are given that converges absolutely, this implies that converges. By definition this means that the sequence of partial sums, , converges. But every convergent sequence is also Cauchy.
Let and let . Since is Cauchy there is an such that for every . We wish to show that converges by showing that it’s Cauchy, that is, that for .
Consider multiplied times. Each of the is < for . This gives us the following inequality when : since . This implies that for every , thus is Cauchy. Thus converges and converges.
Reflection: The meat of this proof is in the Cauchy criterion for series, which allowed us to show the partials of the series converged, giving us that the series itself converges.