Hi
Maybe you can use an "explicit" form of showing by induction that,
given
for every p integer
from to ,
where is a strictly increasing sequence of positive integers
Show that the sequence of positive integers given by is strictly increasing.
I have written down a reductio-ad-absurdum-proof, but it's based largely on tedious reduction of indices in and eventually showing that the argument is proved by the contradiction , and now I'm not all that sure it is correct. Do you have any better ideas?
If you calculate the first few terms of the sequence in terms of , you get
(for n = 2 to 3)
(for n = 4 to 7)
and so on.
The idea is to show that the subsequence where and is strictly increasing, because each term of this subsequence can be written as , and the are strictly increasing.
You also need to show that for each n, , where and .
Ok, I decided to state that is composed of subsequences , where , and now I want to show that every subsequence , as well as that the last element of is less than the first element of so that there is an increase in the "boundary" when the coefficient by changes --and that's what I have most problems with. Could you help me out?