Question 4.T.3: If a series is absolutely convergent, then it is convergent.

Suppose \sum{|x_{n}|} converges. We shall use Cauchy’s criterion to show that \sum{x_{n}} converges. Let ε be an arbitrary positive number. By Theorem 4.2 there a positive integer N such that

⇒ |x_{m+1} + · · · + x_{n}| < ε,

which means \sum{x_{n}} satisfies the Cauchy criterion for series, and is therefore convergent.