David,
Looking further at what you are trying to prove, its clear that, though
you can have geometric sequences over the naturals, this kind of result
will not be obtainable in that context.
As soon as you thing of them as series, things go awry, since non but
the absolutely trivial ones converge.
So if you want to reason about these things then you will need real
numbers and can best work from maths_egs.
However the leap from the kind of reasoning you have been doing to
formal proof in analysis is very great (also, if it matters to you, this
is no longer discreet mathematics!) and I would have thought that
carrying through some arithmetic proofs would be a more manageable
advance for you.
Roger
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