Chapter 1.7, Problem 9P

Prove that an absolutely convergent series ${\displaystyle {\sum }_{n=1}^{\infty }{a}_n}$ an is convergent. Hint: Put $b_n=a_n+\left|a_n\right|$.

Then the b_n are nonnegative, we have $\left|b_n\right|\le 2\left|a_n\right|and{a}_n=b_n-\left|a_n\right|$.