MATH 23a FALL 2001 THEORETICAL LINEAR ALGEBRA AND MULTIVARIABLE CALCULUS Final Version Homework Assignment 9 Due Friday November 30 2001 Please turn in five separate sets labelled A through E 1 Read Sections 1 8 and 2 1 2 2 from Edwards and re read the Appendix 2 A Finish the following proof from class 11 19 using an explicit argument Let V be a real vector space with a convergent sequence vn such that limn vn v and let cn be a convergent sequence of real numbers with limn cn c Show that cn vn converges to cv 3 A Suppose f Rn R and g Rn R are continuous at a and f suppose g a 6 0 Show that is continuous at a g 4 B A set S is called a metric space if there exists a function d S S R called the distance such that i d x y d y x x y S ii d x y d x z d z y x y z S iii d x x 0 x S iv d x y 0 x y S with x 6 y a Let S be any set and define d x x 0 x S and d x y 1 if x 6 y Show that this is a metric space b Suppose S is a metric space with distance d Show that S is also a metric space with new distance given by d0 x y d x y 1 d x y 5 B Consider S R2 with v a b and w c e We define the Memphis metric by d v v 0 for any v and for v 6 w d v w a2 b2 c2 e2 a Show that S is a metric space See problem 4 b Find B 0 c For v 6 0 find B v for various 0 1 6 C Define f 0 1 R as follows 0 if x 6 Q f x 1 if x Q and x q p q in lowest terms a Graph f b Show that f is not continuous at any rational x c Show that f is continuous at any irrational x 7 D Given a set S in a normed vector space we define a point x S to be an interior point if 0 such that B x S We define the interior of S to be the set of interior points and we denote it by S Show that the interior of any set is open 8 D Let V R2 and consider the following subsets A Q Q x y R2 x y Q and B x y R2 x2 y 2 1 For the following recall that S S and S c denote the closure interior see 7 and complement respectively of S a Find A Ac A Ac and A Ac b Find B B c B B c and B B c c Find A B and A B 9 C Let S x sin x1 x 0 R2 Find S 10 E A subset S in a metric space or a normed vector space is called discrete if for every x S there is some 0 such that B x S x that is the only intersection between the ball and the set is the point itself a Show that every f S R is continuous if S is discrete b Show that every closed bounded and discrete set is finite and give examples why each of these three conditions is necessary c Show that Z R is discrete d Show that Z Qp is not discrete 11 E Let Sn be a collection of open sets in a normed vector space and let Tn be a collection of closed sets Show that a S1 S2 is open S b Sn is open c T1 T2 is closed T d Tn is closed 2