At this stage
was shown to be less than or equal to
which is correct but if this is done the conclusion of the proof is:
will increase without bound if
, so this method does not prove the general case that
is bounded. (When
there is a divide-by-zero problem as well.)
A possible although unwieldy solution is to replace k! with something other than
. I propose a complete solution as follows:
(using
as the ceiling function):
therefore:
is a GP, so using
with
as the number of terms and
as the growth per term:
But for all
:
So
therefore:
which is a bound independent of