Convergence of a Cauchy sequence

[SANGHERA.JAS]

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

 

[ArcanaNoir]

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.