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Given norm $$ \| \{ a_n \} \| =\sum_{i=1}^\infty | a_n|,$$ where $E$ is a vector space of absolutely convergent series of real numbers under pointwise operations. How would one prove completeness of this vector space, that every cauchy sequence converges into an absolutely convergent series of real numbers?

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This is Example 8.5. here: http://www.math.psu.edu/wysocki/M403/Notes403_8.pdf (I would have usually added this as a comment, but I am not allowed to...)

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