Convergence of Cauchy Sequences

The Cauchy product of two seriesanddefined aswithconverges towhereandif and only if at least one ofis absolutely convergent.

Proof: Define

For each

DefineSince converges toit is enough to show thatconverges to zero. Choosethen letand There issuch thatimplies thatand forthus for

Also, ifconverges toconverges toconverges tothen

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