Convergence of a Cauchy sequence

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SANGHERA.JAS
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Since I don't know how to use latex I have posed my question in word file.
Yours help is greatly appreciated.
 

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If the author has already shown that the statement is equivalent to the statement including the absolute value of q (alone in the absolute value, nothing else with it) then to show it for q<0 would give the same result, as the absolute value would make q positive.