Absolutely Convergent Series is Convergent/Real Numbers

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Theorem

Let $\displaystyle \sum_{n \mathop = 1}^\infty a_n$ be an absolutely convergent series in $\R$.


Then $\displaystyle \sum_{n \mathop = 1}^\infty a_n$ is convergent.


Proof

By the definition of absolute convergence, we have that $\displaystyle \sum_{n \mathop = 1}^\infty \size {a_n}$ is convergent.

From the Comparison Test we have that $\displaystyle \sum_{n \mathop = 1}^\infty a_n$ converges if and only if:

$\forall n \in \N_{>0}: \size {a_n} \le b_n$

where $\displaystyle \sum_{n \mathop = 1}^\infty b_n$ is convergent.


So substituting $\size {a_n}$ for $b_n$ in the above, the result follows.

$\blacksquare$


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