Hmm.. How about this..
Take as a given that
is an increasing sequence with
. (It is true:
see here.)
For
we know that
but that same sum diverges for any other r.
Let
. Then, by definition of limit, there exists
such that for each
,
. Thus for any
, there are an infinite number of terms such that
. Hence,
is greater than any (monic) convergent geometric series, and is therefore divergent.
QED?