exercise 3

part a

Dv={tvt>0}R2. Consider a sequence of vectors vnR2,n1 such that vnv=0. For example vn=(1,1)v=(1,0) 1 a) Show that the set R2\(n1DEn) is a neighborhood of every point in R2\({0}Dv(n1Dvn)), but is not a neighborhood of any point in {0}Dv(n1Dvn). b) Show that neither n1Dvn nor {0}n1Dvn is a closed subset of R2. c) Show that {0}Dv(n1Dvn) is a closed subset of R2