This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: x k ≤ x k1 , it follows that1 x k ≤ 1 x k1 , and x k +1 = 21 x k ≤ 21 x k1 = x k . Thus the sequence is monotonically decreasing. We next show that it is bounded below. Indeed, we show that x n > 1. We, again, use induction. The base case is given by hypothesis. Assuming that x k > 1, we examine x k +1 . Indeed, x k > 1 gives us that1 x k >1 and x k +1 = 21 x k > 21 = 1 which establishes the bound from below. Since the sequence is monotonically decreasing and bounded below, it must converge. Set L = lim n →∞ x n . Then, we must have that L = 21 L 1 or L 2 = 2 L1 . Thus, L = 1. 2...
View
Full
Document
This note was uploaded on 07/15/2010 for the course MATH 521 taught by Professor Metcalfe during the Spring '10 term at University of North Carolina Wilmington.
 Spring '10
 MetCalfe
 Calculus

Click to edit the document details