# Proof of limit

• Jan 14th 2011, 11:23 AM
mathsohard
Proof of limit
Suppose C > 1 and Sn = C^(1/n). Show that 1 < Sn and Sn+1 < Sn(assume the contrary in each case, and deduce a contraduction). Hense lim n->oo Sn exists. Denote the limit by r. Why is r >= 1? Prove that r = 1 by showing that r > 1 leads to the contradiction that 1/c < = 0.

This is the question, however I don't even understand the question and can't even start on it. HELP !
• Jan 14th 2011, 11:49 AM
HallsofIvy
Sn= C^{1/n} so S1= C, S2= C^{1/2}, S3= C^{1/3}. etc. The first thing you are asked to show is that Sn> 1 for all n. Can you show that?

Then you are asked to show that S(n+1)< Sn, that is, that the sequence is decreasing- and there is a hint that you use proof by contradiction. Suppose that for some n, S(n+1)>= Sn so that C^{1/(n+1)}>= C^{1/n}. Since these are positive numbers, taking the n+ 1 power of each side won't change the direction of the inequality: C>= C^(n+1)/n= C(C^{1/n}. Why is that a contradiction.

Suppose that the limit, r, is less than 1. Let epsilon= 1- r. Then there exist N such that if n> N, |Sn- r|< epsilon (if you don't see why you are in the wrong class) so that Sn- r< 1- r giving Sn< 1, contradicting the very first thing you were to show.
• Jan 14th 2011, 12:37 PM
mathsohard
For the Sn>1 part, since C > 1, nsqrt of C>1 is always bigger than 1 right? Do I need to show why???

and everything else I understood well thanks ^^