This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: R , then inf E is an element of E . Solution . No, e.g., inf(0 , 1) = 0. (d) Let U n be a sequence of open sets in R and let U = ∞ i n =1 U n . Then every point of U is an interior point of U . Solution . No, e.g., U n = (1 /n, 1 /n ). Then U = { } , and 0 is not an interior point of U . (e) Any bounded sequence in R converges. Solution . No, e.g., a n = (1) n . (f) Any Cauchy sequence in a metric space converges. Solution . No, e.g., a n consists of the ±rst n digits of the decimal expansion for √ 2. 1 3. (20 points) Show using the defnition oF convergence that lim n →∞ n 2 n 2 + 1 = 1 . No points will be awarded if the deFnition isn’t used! Solution . Let ǫ > 0. Let N be an integer satisfying N > r (1 /ǫ )1. Then for n ≥ N , we have v v v v n 2 n 2 + 11 v v v v = 1 n 2 + 1 < 1 ( r (1 /ǫ )1) 2 + 1 = ǫ. This gives the desired convergence. 2...
View
Full
Document
This note was uploaded on 03/19/2012 for the course MATH 171a taught by Professor Staff during the Spring '08 term at UCSD.
 Spring '08
 staff
 Math

Click to edit the document details