If then for n=1,2,3,... (easy proof by induction). Put and you see that . The contrapositive version of that is: .
I am trying to prove that for any such that . We have that for any , can be made arbitrarily close to 1.
I'm completing the routine outlined in Rudin's book. So I'm at the step where I need to prove: . And I have this so far:
. Which I expand by the binomial theorem to get:
Where my plan was to drop all but the last two terms by claiming that they are non-negative and therefore: . However to claim they are non-negative I need to use the fact that . Which is the fact I'm trying to prove.
So can anyone see what I'm doing incorrectly at this point? Thanks in advance.
Thank you again for the reply.