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**Is my proof for "Absolute convergence => Convergence" right?**

I had to prove it for my calc test today; my proof looks different than the one my prof posted as the answer; it's also way shorter. He used the one from the book but I didn't memorize it so I used this. Is it right?

## Homework Statement

Prove that Absolute Convergence implies convergence

## The Attempt at a Solution

Notation:

1. I'm using the notation

**a**n means a series. [As in:

**a**n =

**a**(1)+

**a**(2)+

**a**(3)...+

**a**(n) and n goes to infinity]

2. |

**a**n| means the absolute value of |

**a**n|.

***

I. |(-1)^n

**a**n| =

**a**n.

II. Suppose

**a**n converges. i.e:

**a**n = C

III.

**a**n converges implies that

**a**(n+1) <

**a**(n)

IV. (III.) implies (-1)^n

**a**n converges.

QED.

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