I'm having a lot of trouble showing that that the
I figure it has to be done by induction, so if I wanted to show that tends to infinity faster then I would have:
which is obviously true
so assume that
then I would have where the inequality holds only if tends to infinity at a faster rate then
of we have which will follow if you show that this is clear for for let and apply L'Hospital rule times to get the result.
I will be explicit in my hint, I guess you did not see what I was trying to say. Let and . Pick so that . If we can prove that for sufficiently large the proof will follow because . However, we know that . However, if we have . Thus, we have shown that eventually overtakes and so it overtakes .